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Doctoral Thesis

Anti-self-dual instantons with Lagrangian boundary conditions

Author(s): Wehrheim, Katrin

Publication Date: 2002

Permanent Link: https://doi.org/10.3929/ethz-a-004446022

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ETH Library

Diss. ETH No. 14747

Anti-self-dual instantons with

Lagrangian boundaryconditions

A dissertation submitted to the

SWISS FEDERAL INSTITUTE OF TECHNOLOGY

ZÜRICH

for the degree of

Doctor of Mathematics

presented by

KATRIN WEHRHEIM

Dipl. Math. ETH Zürich

born June 11, 1974

citizen of the Federal Republic of Germany

accepted on the recommendation of

Prof. Dr. D. Salamon, examiner

Prof. Dr. D. McDuff, co-examiner

Prof. Dr. E. Zehnder, co-examiner

2002

Abstract

We consider a nonlocal boundary condition for anti-self-dual instantons on

four-manifolds with a space-time splitting of the boundary. This boundarycondition naturally arises from making the Chern-Simons functional on a

three-manifold with boundary closed: The restriction of the instanton to each

time-slice of the boundary is required to lie in a Lagrangian submanifold of

the moduli space of flat connections.

We establish the fundamental elliptic regularity and compactness prop¬

erties of this boundary value problem. Firstly, every weak solution is gauge

equivalent to a smooth solution. Secondly, all closed subsets of the moduli

space of solutions with an i7-bound on the curvature for p > 2 are compact.

We moreover establish the Fredholm property of the linearized operator of

this boundary value problem on compact four-manifolds. The proofs are

based on a decomposition of the instantons near the boundary. Due to the

global nature of the boundary condition the crucial regularity of one of these

components has to be established by studying Cauchy-Riemann equationswith totally real boundary conditions for functions with values in a complexBanach space.

These results provide the basic analytic set-up for the definition of a Floer

homology for pairs consisting of a compact three-manifold with boundaryand a Lagrangian submanifold in the moduli space of flat connections over

the boundary. Such a Floer homology lies at the center of the program for

the proof of the Atiyah-Floer conjecture by Salamon. The program aims

to use this Floer homology as intermediate step for the conjectured natural

isomorphism between the instanton Floer homology of a homology three-

sphere and the symplectic Floer homology of two Lagrangian submanifolds

in a moduli space of flat connections arising from a Heegard splitting of the

homology three-sphere. These isomorphisms should result from adiabatic

limits of the boundary value problem that is studied in this thesis.

i

ii

ZusammenfassungWir betrachten eine nichtlokale Randbedingung für antiselbstduale Instan¬

tone auf Viermannigfaltigkeiten, deren Rand eine Raum-Zeit-Aufspaltung

trägt. Diese Randbedingung ist die natürliche Bedingung, durch die das

Chern-Simons-Funktional einer Dreimannigfaltigkeit mit Rand geschlossenwird: Die Einschränkung des Instantons auf jede Rand-Teilfläche konstanter

Zeit liegt in einer Lagrangeschen Untermannigfaltigkeit des Modulraumes der

flachen Zusammenhänge.Wir zeigen die grundlegenden elliptischen Regularitäts- und Kompakt¬

heits-Eigenschaften dieses Randwertproblemes. Jede schwache Lösung ist

eichäquivalent zu einer starken Lösung. Weiterhin ist jede abgeschlossene

Teilmenge des Modulraumes der Lösungen, die eine i7-Schranke für die

Krümmung mit p > 2 hat, kompakt. Zudem zeigen wir die Fredholm-

Eigenschaft des linearisierten Operators dieses Randwertproblemes auf kom¬

pakten Viermannigfaltigkeiten. Die Beweise basieren auf einer Zerlegung des

Instantons in der Nähe des Randes. Die Regularität einer dieser Komponen¬ten macht es wegen der globalen Natur der Randbedingung nötig, Cauchy-

Riemann-Gleichungen mit total reellen Randbedingungen für Funktionen mit

Werten in einem komplexen Banachraum zu studieren.

Diese Resultate legen die analytischen Grundlagen für die Definition einer

Floer-Homologie für Paare bestehend aus einer kompakten Dreimannigfaltig¬keit mit Rand und einer Lagrangeschen Untermannigfaltigkeit des Modul¬

raumes der flachen Zusammenhänge über dem Rand. Eine solche Floer-

Homologie liegt im Zentrum des Beweisprogrammes von Salamon für die

Atiyah-Floer-Vermutung. In diesem dient die erwähnte Floer-Homologie als

Zwischenschritt in der Konstruktion des vermuteten natürlichen Isomorphis¬mus' zwischen der Instanton-Floer-Homology einer Homologie-Dreisphäreund der symplektischen Floer-Homologie zweier Lagrangescher Unterman¬

nigfaltigkeiten eines Modulraumes flacher Zusammenhänge, die aus einer

Heegard-Zerlegung der Homologie-Dreisphäre resultieren. Die zwei Isomor¬

phismen in diesem Programm sollten sich aus adiabatischen Limites des in

dieser Doktorarbeit betrachteten Randwertproblemes ergeben.

iii

iv

Contents

Abstract i

Zusammenfassung iii

1 Introduction 1

2 Floer homology and the Atiyah-Floer conjecture 17

3 Weakly flat connections 31

4 Lagrangians in the space of connections 37

5 Cauchy-Riemann equations in Banach spaces 47

6 Regularity and compactness 67

7 Fredholm theory 95

8 Bubbling 115

A Gauge theory 129

Bibliography 141

Acknowledgements 145

Curriculum Vitae 147

CONTENTS

Chapter 1

Introduction

We begin by giving a brief introduction to the gauge theory of connections

on principal bundles.

Let G be a compact Lie group. A principal G-bundle is a fibre bundle

Ti : P —> M with a free, transitive action of G on the fibres tt~1(x) = G.

The vertical subbundle V = ker(d7r) C TP is then given by the tangencies

^p = {p£ | Ç ^ fl} to the orbits. Here g denotes the Lie algebra of G and

p i—) p^ is the infinitesimal action of £ G g. Now a horizontal distribution

H c TP is an equivariant choice of complements TpP = Vp © Hp.Consider a trivialization P\u = U x G over U C M. Then horizontal

distributions over U can be identified with g-valued 1-forms A G r21(C7;fl)via the following identity: For every p = (x,g) G U x G one identifies

TXU x T9G ^ TPP, then

Hp = {(Y,-gAx(Y))\YeTxU}.

Here the 1-form A determines the deviation from the natural horizontal dis¬

tribution H(g>x) = TXU x {0}. On a nontrivial bundle there is no such natural

choice, but still the horizontal distributions can be identified with the kernels

Hp = kevAp of certain 1-forms A e A(P) called connections. The space of

smooth connections A(P) C ^X(P; g) consists of equivariant g-valued 1-forms

with fixed values on the vertical bundle.

The automorphisms of a principal G-bundle are of the form p i—> pu(p)with an equivariant smooth map u : P —> G called a gauge transformation.

These form the gauge group G(P) which acts on the space of connections:

The pullback of a connection A e A(P) under a G-bundle automorphism

given by u G G(P) is denoted by u*A and is called gauge equivalent to A.

1

2 CHAPTER 1. INTRODUCTION

For simplicity of notation now consider the trivial G-bundle P = M x G

over a manifold M. Then the space of connections is A(M) := fl1(M; g), the

gauge group can be viewed as G(M) := C°°(M, G), and the action of a gauge

transformation u G G(M) on a connection A G *4(M) is given by

u*A = u~lAu + u~1du.

A connection A G A(M) is called flat if the associated horizontal distribution

is locally integrable. This is equivalent to d^ o d^ = 0 for the exterior

differential d^ : flk(M;g) -> Qk+1(M;g), dAu) = du + [AAuj] associated with

the connection. One has d^d^u; = [Fa A u] for all lu G Qk(M; g), where

FA = dA + l[AAA] G tf(X;g)

is called the curvature of A. So a connection is flat if and only if its curvature

vanishes. Now the Yang-Mills energy of a connection,

yM(A) = f \FA\2,Jm

is a gauge invariant measure for the nonintegrability of the horizontal dis¬

tribution. The extrema of the Yang-Mills functional are the solutions of the

Yang-Mills equation d^F^ = 0. Note that the definition of the Yang-Millsfunctional and more generally of the L2-inner product of g-valued differen¬

tial forms uses a metric on M as well as a G-invariant metric on G. The

coderivative d^ then is defined as the formal L2-adjoint of d^. (More details

about gauge theory and these notations can be found in appendix A.)In the case of a manifold with boundary, the extrema of yAi moreover

satisfy the boundary condition *Fa\om = 0. So we call A G A(M) a Yang-Mills connection if it satisfies the boundary value problem

f d*AFA = 0,

\*FA\dM = 0.

Extrema of yAi actually are weak Yang-Mills connections, that is they sat¬

isfy the weak equation

[ (FA, dAß) = 0 yßeQ\M;g),Jm

where ß runs through all smooth 1-forms. However, one has the following

regularity result, which we state in the case of a G-bundle (not necessarily

trivial) over a compact 4-manifold M. Here we use the notation Ak,p(M)

3

and Gk,p(M) for the VF"*'p-Sobolev completions of the space of connections

and the gauge group.

Theorem (Regularity for Yang-Mills connections)Let p > 2. Then for every weak Yang-Mills connection A G A1,P(M) there

exists a gauge transformation u G G2,P{M) such that u*A is smooth.

An elementary observation in gauge theory is that the moduli space of

gauge equivalence classes of flat connections on a G-bundle over a compact

manifold M is compact in the C°°-topology. This is obvious from the fact

that the gauge equivalence classes of flat connections over M are in one-

to-one correspondence with the conjugacy classes of representations of the

fundamental group of M, cf. theorem A.2. The Uhlenbeck compactness the¬

orems are a remarkable generalization of this result. We again state them

in the case of a (not necessarily trivial) G-bundle over a compact 4-manifold

M. Proofs of these and the above theorem can be found in [U2],[DK], or

[We] (explicitly containing the case of manifolds with boundary).

Theorem (Weak Uhlenbeck Compactness)Let {Av)vÇ^ c A1,P(M) be a sequence of connections and suppose that ||Fa"|Ipis uniformly bounded for some p > 2. Then there exists a subsequence (againdenoted (A")^^) and a sequence of gauge transformations uv G G2,P(M) such

that uv *Av converges weakly in A1,P{M).

Theorem (Strong Uhlenbeck Compactness)Let {Av)ve^ C A1,P(M) be a sequence of weak Yang-Mills connections and

suppose that ||Pa"||p is uniformly bounded for some p > 2. Then there exists

a subsequence (again denoted (Au)ue^) and a sequence of gauge transforma¬tions uu G G2'P{M) such that uv *Av converges uniformly with all derivatives

to a smooth connection A G A(M).

An important application of Uhlenbeck's theorems is the compactifica-tion of the moduli space of (gauge equivalence classes of) anti-self-dual in¬

stantons over a four-manifold. These compactified moduli spaces are the

central ingredients in the construction of the Donaldson invariants of smooth

four-manifolds [D2] and of the instanton Floer homology groups of three-

manifolds [Fl]. Anti-self-dual instantons are special first order solutions of

the Yang-Mills equation described in the following.

4 CHAPTER 1. INTRODUCTION

Let M be a closed oriented 4-manifold, choose a metric, and denote the

associated Hodge operator by *. Then the curvature Fa = FA + FA splitsinto a self-dual and an anti-self-dual part, FA = \{Fa ± *Pa)- Now an

elementary calculation shows that the Yang-Mills energy can be rewritten as

yM(A) = - [ (FAAFA) + 2 / \F+\2. (1.1)Jm Jm

The first term on the right hand side is a topological invariant of the bundle

P —> M. (For example, in the case G = SU (2) this invariant equals 87r2c2(P),where C2(P) is the second Chern number and the inner product on su(2) is

given by the negative trace of the product of the matrices.) If this invariant

is nonnegative, then the minima of the Yang-Mills functional are exactly the

anti-self-dual instantons, i.e. connections A G A(M) with

FA + *FA = 0.

The formula (1.1) shows that a sequence of anti-self-dual connections on a

given bundle automatically has an L2-bound on the curvature. Uhlenbeck's

compactness theorem however does not extend to the case p = 2 due to the

bubbling phenomenon:The conformally invariant Yang-Mills energy yAi(A) = ||Fa|||2 can con¬

centrate at single points. If this happens at an interior point, then rescalingnear that point yields a sequence of connections on balls of increasing radii

whose limit modulo gauge is a nontrivial anti-self-dual instanton that ex¬

tends to S4. Its energy equals some positive constant times a characteristic

number of the bundle. (This number is nonzero due to the nontrivialityof the instanton; for an SU(2)-bundle it is the second Chern number.) So

on a closed manifold and for a suitably chosen subsequence this bubbling

only occurs at finitely many points. On the complement of these points,Uhlenbeck's compactness theorems ensure C°°-convergence on all compact

subsets. Now Uhlenbeck's removable singularity theorem [Ul] guarantees

that the limit connection extends over the 4-manifold (to a connection on a

bundle of lower characteristic number). This leads to a compactification of

the moduli space of anti-self-dual instantons. In the case of simply connected

4-manifolds with negative definite intersection forms Donaldson used these

compactified moduli spaces to prove his famous theorem about the diagoniz-

ability of intersection forms [Dl].

5

Now in the case of a 4-manifold with boundary note that the boundarycondition *Fa\om = 0 for anti-self-dual instantons implies that the curvature

vanishes altogether at the boundary. This is an overdetermined boundaryvalue problem comparable to Dirichlet boundary conditions for holomorphic

maps.x As in the latter case it is natural to consider weaker Lagrangian

boundary conditions.

More precisely, a natural boundary condition for holomorphic maps with

values in an almost complex manifold (X, J) is to take boundary values in

a totally real submanifold. Recall that J G End(TM) is called an almost

complex structure if J2 = — t. A submanifold L c X is called totally real if

TXL® JTXL = TXX for all x G L. So essentially, it suffices to have Dirichlet

boundary conditions for half of the components of the holomorphic map.

Special cases of almost complex manifolds are symplectic manifolds (X, u)with ^-compatible almost complex structures. In this symplectic case the

Lagrangian submanifolds are examples of totally real submanifolds.

We consider a version of such Lagrangian boundary conditions for anti-

self-dual instantons and prove that they suffice to obtain the analogue of the

above regularity and compactness results for Yang-Mills connections.

For that purpose we consider oriented 4-manifolds X with a space-time

splitting of the boundary, i.e. each connected component of dX is diffeomor-

phic to S x E, where S is a 1-manifold and E is a closed Riemann surface.

We shall study a boundary value problem associated to a gauge invariant

Lagrangian submanifold £ of the space of flat connections on E: The re¬

striction of the anti-self-dual instanton to each time-slice of the boundaryis required to belong to C. This boundary condition arises naturally from

examining the Chern-Simons functional on a 3-manifold Y with boundaryE. Namely, the Langrangian boundary condition renders the Chern-Simons

1 Consider for example the abelian case G = S1, then the anti-self-duality equationis dA = 0. The gauge freedom is eliminated by going to a local slice, dM = 0 and

*y4|,9m = 0. This already gives the elliptic boundary value problem AA = 0 with Dirich¬

let boundary conditions for the normal component of A and Neumann boundary con¬

ditions for the components of A in directions tangential to the boundary. Indeed, let

A = Aodxç + X]ü=i Atdxi, where Xq is the coordinate normal to the boundary dM and

the x, are coordintes of dM. Then the local slice boundary condition is j4o|x0=o = 0

and the boundary condition *Fa\bm = 0 gives doAt\xo=Q = 0 for all î = l,...,n — 1.

So the solutions of this boundary value problem are already unique (up to a constant

in the tangential components). But now the anti-self-duality equation gives the addi¬

tional boundary condition Fa\bm = 0, in above coordinates dlAJ\xo=o — dJAl\xo=o for all

i,jE{l,...,n — 1}. This makes the boundary value problem overdetermined.

6 CHAPTER 1. INTRODUCTION

1-form on the space of connections closed, see [Sal]. The resulting gradientflow equation leads to the boundary value problem studied in this thesis (forthe case X = M x Y), as will be explained in chapter 2. Our main results

establish the basic regularity and compactness properties as well as the Fred-

holm theory, the latter for the compact model case X = S1 x Y.

Boundary value problems for Yang-Mills connections were already con¬

sidered by Donaldson in [D3]. He studies the Hermitian Yang-Mills equationfor connections induced by Hermitian metrics on holomorphic bundles over

a compact Kahler manifold Z with boundary. Here the unique solubilityof the Dirichlet problem (prescribing the metric over the boundary) leads

to an identification between framed holomorphic bundles over Z (meaninga holomorphic bundle with a fixed trivialization over dZ) and Hermitian

Yang-Mills connections over Z. In particular, when Z has complex dimen¬

sion 1 and boundary dZ = S1, this links loop groups to moduli spaces of flat

connections over Z. This observation suggests an alternative approach to

Atiyah's [Al] correspondence between holomorphic curves in the loop group

of a compact Lie group G and anti-self-dual instantons on G-bundles over

the 4-sphere. The correspondence might be established via an adiabatic limit

relating holomorphic spheres in the moduli space of flat connections over the

disc to anti-self-dual instantons over the product (of sphere and disc). Our

motivation for studying the present boundary value problem lies more in

the direction of another such correspondence between holomorphic curves in

moduli spaces of flat connections and anti-self-dual instantons - the Atiyah-Floer conjecture for Heegard splittings of a homology-3-sphere.

A Heegard splitting Y = Y0 Ue Y\ of a closed 3-manifold F is a decom¬

position into two handlebodies lo and Y\ with common boundary E. (Ahandlebody is a 3-ball with a finite number of solid handles attached.) It

gives rise to two Floer homologies, i.e. generalized Morse homologies. Firstly,the moduli space ME of gauge equivalence classes of flat connections on the

trivial SU(2)-bundle over E is a symplectic manifold (with singularities) and

the moduli spaces LYt of flat connections over E that extend to Yt are La¬

grangian submanifolds of Ms (see chapter 4). The symplectic Floer homology

HF^ymp(Mg, LYo, LYl) is now generated by the intersection points of the La¬

grangian submanifolds (as critical points of a generalized Morse theory). The

critical points in the case of the instanton Floer homology HFJKnst(F) are the

flat connections over Y.

7

Conjecture (Atiyah, Floer) Let Y = Y0 Us Y\ be a Heegard splitting ofa homology 3-sphere. Let LYt be the Lagrangian submanifolds of the moduli

space Ms of flat SXJ (2)-connections over E given by the flat connections over

E that extend to Yt. Then there exists a natural isomorphism between the

instanton and symplectic Floer homologies

HFmSt(y) ^ HFfmP(ME,Lyo,Lyi).

The program for the proof by Salamon, [Sal], is to define the instanton

Floer homology HFst (Y, L) for 3-manifolds with boundary dY = E using

boundary conditions associated with a Lagrangian submanifold L C Ms.

Then the conjectured isomorphism might be established in two steps via the

intermediate HFst([0,1] x E,Ly0 x LYl) by adiabatic limit type arguments

similar to [DS2].Fukaya was the first to suggest the use of Lagrangian boundary condi¬

tions in order to define a Floer homology for 3-manifolds Y with boundary,

[Ful]. He studies a slightly different equation, involving a degeneration of

the metric in the anti-self-duality equation, and uses SO(3)-bundles that are

nontrivial over the boundary dY. Now there are interesting examples, where

one has to work with the trivial bundle. For example, on a handlebody Y

there exists no nontrivial G-bundle for connected G. So if one considers any

3-manifold Y with the Lagrangian submanifold CYi, the space of flat connec¬

tions on dY = dY' that extend over a handlebody Y', then one also deals

with the trivial bundle. So if one wants to use Floer homology on 3-manifolds

with boundary to prove the Atiyah-Floer conjecture, then it is crucial to ex¬

tend this construction to the case of trivial SU(2)-bundles. There are two

approaches that suggest themselves for such a generalization. One would be

the attempt to extend Fukaya's construction to the case of trivial bundles,and another would be to follow the alternative construction outlined in [Sal].The present thesis follows the second route and sets up the basic analysis for

this theory. We will only consider trivial G-bundles. However, our main

theorems A, B, and C below generalize directly to nontrivial bundles - justthe notation becomes more cumbersome.

8 CHAPTER 1. INTRODUCTION

The main results

We consider the following class of Riemannian 4-manifolds. Here and through¬out all Riemann surfaces are closed 2-dimensional manifolds. Moreover, un¬

less otherwise mentioned, all manifolds are allowed to have a smooth bound¬

ary.

Definition 1.1 A 4-manifold with a boundary space-time splittingis a pair (X, r) with the following properties:

(i) X is an oriented 4-manifold which can be exhausted by a nested sequence

of compact deformation retracts.

(ii) t = (ri,...,rn) is an n-tuple of embeddings r% : S% x E? —> X with

disjoint images, where Ej is a Riemann surface and St is either an

open interval in M or is equal to S1 = M/Z.

(Hi) The boundary OX is the union

n

dX = \Jrl(SixEi).i=\

Definition 1.2 Let (X, r) be a 4-manifold with a boundary space-time split¬

ting. A Riemannian metric g on X is called compatible with r if for each

i = 1,... n there exists a neighbourhood Ut C St x [0, oo) of St x {0} and an

extension of rz to an embedding fz : Ut x Ej —> X such that

f*g = ds2 + dt2 + gStt.

Here gs<t is a smooth family of metrics on E, and we denote by s the coordinate

on St and by t the coordinate on [0, oo).We call a triple (X, r, g) with these properties a Riemannian 4-manifold

with a boundary space-time splitting.

Remark 1.3 In definition 1.2 the extended embeddings ft are uniquely de¬

termined by the metric as follows. The restriction fz\t=o = t% to the boundaryis prescribed, and the paths t *-$ fz(s,t, z) are normal geodesies.

9

Example 1.4 Let X := IR x Y, where Y is a compact oriented 3-manifold

with boundary dY = E, and let r : R x E —> X be the obvious inclusion.

Given any two metrics g_ and g+ on Y there exists a metric g on X such

that g = ds2 + g_ for s < — 1, g = ds2 + g+ for s > 1, and (X, r, g) satisfies

the conditions of definition 1.2. The metric g cannot necessarily be chosen

in the form ds2 + gs (one has to homotop the embeddings and the metrics).

Now let (X, r, g) be a Riemannian 4-manifold with boundary space-time

splitting and consider a trivial G-bundle over X for a compact Lie group

G. Let p > 2, then for each i = 1,... ,n the Banach space of connections

^4°'p(Ej) carries the symplectic form u(a,ß) = Js (a A ß). We fix an n-

tuple £ = (£i,..., £n) of Lagrangian submanifolds Ct C A°'p(T,t) that are

contained in the space of flat connections and that are gauge invariant,

Ac^(E0 and «*A = A Vueglj,(zt).

Here >4fl^.(Ej) is the space of weakly flat I^-connections on Ej (see chapter 3).A submanifold £ of a symplectic Banach space (Z, uj) is called Lagrangianif it is isotropic, i.e. u\c = 0, and is of maximal dimension. To make the

latter precise in this infinite dimensional setting, choose a complex structure

J End Z that is compatible with uj (i.e. o>(-, J-) defines a metric on Z).The condition of maximal dimension is now phrased as the topological sum

Tz£ © JT^£ = TZZ \/z e C.

This condition is independent of the choice of J since the space of compatible

complex structures on (Z,u) is connected [MSI, Proposition 2.48]. Next, the

assumptions on the Ct ensure that the quotients

L% := Ct/G(^)CAZ(^)/G1'p(^) = MSi

are (singular) Lagrangian submanifolds in the (singular) moduli space of

flat connections, cf. chapter 4. We consider the following boundary value

problem for connections A G A^^X)

r *fa + fa = o,(l 2)

\ r*A\{s}x^ e A Vs eSt, i = l,...,n.

Observe that the boundary condition is meaningful since for every neigh¬bourhood WxSofa boundary component one has the continuous embedding

WX'P{U x E) C W^iU, LP{E)) ^ C°{U, L"(E)). The first nontrivial observa¬

tion is that every connection in L% is gauge equivalent to a smooth connection

10 CHAPTER 1. INTRODUCTION

on Ej and hence L% fl *4(E) is dense in £,, see theorem 3.1. Moreover, ev¬

ery Il/^-connection on X satisfying the boundary condition in (1.2) can be

locally approximated by smooth connections satisfying the same boundary

condition, see corollary 4.2.

Note that the present boundary value problem is a first order equationwith first order boundary conditions (flatness in each time-slice). Moreover,the boundary conditions contain some crucial nonlocal (i.e. Lagrangian) in¬

formation. We moreover emphasize that while Cz is a smooth submanifold

of *4°'p(Ej), the quotient Cl/G1,p(£'i) is n°t required to be a smooth subman¬

ifold of the moduli space MSi := AQ^t(T,A/G1,p(T,t). For example, Lt could

be the set of flat connections on E4 that extend to flat connections over a

handlebody with boundary E,, see lemma 4.3. To overcome the difficulties

arising from the singularities in the quotient, we will work with the (smooth)quotient by the based gauge group.

We will not be interested in existence results for the present boundaryvalue problem but in its elliptic properties. The following two theorems

are the main regularity and compactness results for the solutions of (1.2).The regularity theorem is the analogue of the regularity theorem for Yang-Mills connections stated above. The compactness theorem deals with con¬

nections satisfying uniform i7-bounds on the curvature and thus extends

Uhlenbeck's strong compactness theorem for anti-self-dual instantons to the

present boundary value problem.

Theorem A (Regularity)Let p > 2. Then every solution A G AX^C{X) of (1.2) is gauge equivalent to

a smooth solution, i.e. there exists a gauge transformation u G G\oc(X) such

that u*A G A(X) is smooth.

Theorem B (Compactness)Letp > 2 and let gv be a sequence of metrics compatible with r that uniformly

converges with all derivatives on every compact set to a smooth metric. Sup¬

pose that Au G A^Z{X) is a sequence of solutions of (1.2) with respect to

the metrics gv such that for every compact subset K C X there is a uniformbound on the curvature ||-Fa"IUp(#)- Then there exists a subsequence (againdenoted Av) and a sequence of gauge transformations uv G G\£{X) such that

uv *Av converges uniformly with all derivatives on every compact set to a

smooth connection A G A(X).

11

The difficulty of these results lies in the global nature of the boundarycondition. This makes it impossible to directly generalize the proof of the

regularity and compactness theorems for Yang-Mills connections, where one

chooses suitable local gauges, obtains the higher regularity and estimates

from an elliptic boundary value problem, and then patches the gauges to¬

gether. With our global Lagrangian boundary condition one cannot obtain

local regularity results.

However, an approach by Salamon can be generalized to manifolds with

boundary. One first uses Uhlenbeck's weak compactness theorem to find a

weakly W^-convergent subsequence. The limit then serves as reference con¬

nection with respect to which a further subsequence can be put into relative

Coulomb gauge globally (on large compact sets). Then one has to establish

elliptic estimates and regularity results for the given boundary value problem

together with the relative Coulomb gauge equations. All the general tools for

this approach are established in [We]: We give a detailed proof of the general¬ization of weak Uhlenbeck compactness to manifolds with boundary that are

exhausted by compact deformation retracts. We also generalize Salamon's

subtle local slice theorem to compact manifolds with boundary. Moreover,

we give a precise formulation of the procedure by Donaldson and Kronheimer

that allows to extend regularity and compactness results on compact defor¬

mation retracts of noncompact manifolds to the full manifolds.

In this thesis, we concentrate on the last step - the higher regularity and

estimates for the boundary value problem in relative Coulomb gauge. Here

the crucial point is to establish the higher regularity or estimates for the

E-component of the connections in a neighbourhood U x E of a boundary

component. The global nature of the boundary condition forces us to deal

with a Cauchy-Riemann equation on U with values in the Banach space

*4°'P(E) and with Lagrangian boundary conditions.

The case 2 < p < 4, when W^-functions are not automatically continu¬

ous, poses some special difficulties in this last step. Firstly, in order to obtain

regularity results from the Cauchy-Riemann equation, one has to straightenout the Lagrangian submanifold by going to suitable coordinates. This re¬

quires a C°-convergence of the connections, which in case p > 4 is given by

a standard Sobolev embedding. In case p > 2 one still obtains a special

compact embedding W1,P(IA xS)4 C°(U, I^(E)) that suits our purposes.

Secondly, the straightening of the Lagrangian introduces a nonlinearity in the

Cauchy-Riemann equation that already poses some problems in case p > 4.

12 CHAPTER 1. INTRODUCTION

In case p < 4 this forces us to deal with the Cauchy-Riemann equation with

values in an L2-Hilbert space and then use some interpolation inequalitiesfor Sobolev norms.

Now as a first step towards the definition of a Floer homology for 3-

manifolds with boundary consider the moduli space of finite energy solutions

of (1.2),

M(C) := {A G AocPO I A satisfies (1.2),yM(A) < oo}/G?£(X).

Theorem A implies that for every equivalence class [A] G Ai(C) one can

find a smooth representative A G A(X). Theorem B is one step towards a

compactness result for Ai{C): Every closed subset of Ai(C) with a uniform

i7-bound for the curvature is compact. In addition, theorem B allows the

metric to vary, which is relevant for the metric-independence of the Floer

homology.Our third main result is a step towards proving that the moduli space

Ai{C) of solutions of (1.2) is a manifold whose components have finite (butpossibly different) dimensions. This also exemplifies our hope that the further

analytical details of Floer theory will work out along the usual lines once the

right analytic setup has been found in the proof of theorems A and B.

In the context of Floer homology and in Floer-Donaldson theory it is

important to consider 4-manifolds with cylindrical ends. This requires an

analysis of the asymptotic behaviour which will be carried out elsewhere. We

shall restrict the discussion of the Fredholm theory to the compact case. The

crucial point is the behaviour of the linearized operator near the boundary;in the interior we are dealing with the usual anti-self-duality equation. Hence

it suffices to consider the following model case. Let F be a compact oriented

3-manifold with boundary dY = E and suppose that (gs)ses1 is a smooth

family of metrics on Y such that

X = S1 x Y, riS'xS^I, g = ds2 + gs

satisfy the assumptions of definition 1.2. Here the space-time splitting r

of the boundary is the obvious inclusion r : S1 x E <—> dX = S1 x E,where E = U=1 Ej might be a disjoint union of Riemann surfaces Ej. The

above n-tuple of Lagrangian submanifolds L% C A0,p(T,t) then defines a gauge

invariant Lagrangian submanifold £ := £i x... x£n of the symplectic Banach

space ^°'P(E) = A°'p{^i) x...

x A°'p{Zn) such that £ c ^(E).

13

In order to linearize the boundary value problem (1.2) together with the

local slice condition, fix a smooth connection A + $ds G A(SX x Y) such

that As := A(s)\dY e C for all s G S1. Here $ G C0O(S'1 x Y,g) and

A G C0O(S'1 x Y,T*Y <g> 0) is an S^-family of 1-forms on y (not a 1-form

on X as previously). Now let EA be the space of S^-families of 1-forms

a G W1,P(S1 x Y, T*Y ® g) that satisfy the boundary conditions

*a(s)\dY = 0 and Q>(s)\oY G T^s£ for all s G S1.

Then the linearized operator

D{A,*) E1/ x W^iS1 x Y,g) — L^S1 x F,T*F ® 0) x ^(S1 x Y,g)

is given with Vs = ds + [$, •] by

A'*.*) (a' <P) = (Vsa - dAy + *dAa , Vsy> - d*Aa).

The second component of this operator is —d*A+^ds(a + (pds), and the first

boundary condition is *(a + (fds)\ox = 0, corresponding to the choice of a

local slice at A + $ds. In the first component of D(a,*) we have used the

global space-time splitting of the metric on S1 x Y to identify the self-dual

2-forms *7S—

7S A ds with families 7S of 1-forms on Y. The vanishing of this

component is equivalent to the linearization dA+^ds(a + (pds) = 0 of the anti-

self-duality equation (see chapter 7). Furthermore, the boundary condition

a(s)\dY T^s£ is the linearization of the Lagrangian boundary condition in

the boundary value problem (1.2).

Theorem C (Fredholm)Let Y be a compact oriented 3-manifold with boundary dY = E and let

S1 x Y be equipped with a product metric ds2 + gs that is compatible with

t : S1 xS4 S1 xY. LetA + $dse A{SX x Y) such that A(s)\dY G £ forall s G S1. Then the following holds for all p > 2.

(i) D(A,<s>) is Fredholm.

(ii) There is a constant C such that for all a G EA and (p G W1,P{S1 x Y, g)

\\(a, (p)\\wi,P < C(||D(Ai$)(a,^)||Lp + ||(a,^)||LP).

14 CHAPTER 1. INTRODUCTION

(m) Let q > p* such that ç / 2. Suppose that ß G Lq{S1 x Y,T*Y <g> 0),£ G Lq(S1 x Y,g), and assume that there exists a constant C such that

for all a G E1/ and tp G W1'P{S1 x Y, g)

[ (AWa^M/U))Js1xY

< C\\(a, L«

Then m fact ß G W1'^1 x Y, T*Y <g> 0) and ( G W1'^1 xY,g) .

Here and throughout we use the notation - + 4r = 1 for the conjugate ex¬

ponent p* of p. The above inner product ( -,• ) is the pointwise inner product

in T*Y ®0X0. The reason for our assumption q ^ 2 in theorem C (iii) is a

technical problem in dealing with the singularities of £/G1,p(E)- We resolve

these singularities by dividing only by the based gauge group. This leads

to coordinates of LP(E,T*E <g> 0) in a Banach space that comprises based

Sobolev spaces W^,p(T,,g) of functions vanishing at a fixed-point zeE. So

these coordinates that straighten out T£ along A\sixdY are welldefined onlyfor p > 2. Now in order to prove the regularity claimed in theorem C (iii) wehave to use such coordinates either for ß or for the test 1-forms a, i.e. we have

to assume that either q > 2 or q* > 2. This is completely sufficient for our

purposes- concluding a higher regularity of elements of the cokernel. This

will be done via an iteration of theorem C (iii) that can always be chosen

such as to jump across q = 2. However, we believe that the use of different

coordinates should permit to extend this result.

Conjecture Theorem C (in) continues to hold for q = 2.

One indication for this conjecture is that the L2-estimate in theorem C (ii)is true (for LU^-regular a and cf> with p > 2), as will be shown in chapter 7.

This L2-estimate can be proven by a much more elementary method than the

general Lp-regularity and -estimates. In fact, it was already stated in [Sal]as an indication for the wellposedness of the boundary value problem (1.2).

Besides the Lp-compactness and Fredholm theory, the construction of an

instanton Floer homology for 3-manifolds with boundary moreover requires

an analysis of the bubbling at the boundary for sequences of solutions of

(1.2) with bounded energy. Here the standard rescaling technique runs into

problems since the local rescaling fails to capture the full information of the

boundary condition. The flatness of the connection on each time-slice of the

15

boundary is a local condition, but the Lagrangian condition is global, i.e.

it can only be stated for a connection on the full time-slice. The rescaling,

however, cuts out small balls in each time-slice. In [Sal] Salamon expected to

obtain anti-self-dual instantons on the half space bubbling off at the bound¬

ary, and he conjectured a quantization of their energy. The convergence of

the rescaled connections to such instantons on the half space, however, would

require some additional information (coming from the Lagrangian boundary

condition) on the local behaviour of the finite energy solutions of (1.2) near

the boundary.

Alternatively, the analytic framework of the regularity and compact¬

ness results obtained in this thesis suggests a global treatment of the time-

slices. This might lead to holomorphic discs in the space of connections

with Lagrangian boundary conditions, that would bubble off as a result of

2-dimensional rescaling only in the time- and interior direction (preservingthe full time-slices). In this thesis we will give some partial results about the

possible bubbling phenomena at the boundary.

Outline

This thesis is organized as follows. We give a short introduction to Floer

homology and the Atiyah-Floer conjecture in chapter 2. We moreover explainin more detail the program for the proof by Salamon and the motivation for

the boundary value problem (1.2).In chapter 3, we introduce the notion of a weakly flat connection on a

general closed manifold. In theorem 3.1, we prove that weakly flat connec¬

tions are gauge equivalent to smooth connections that are flat in the usual

sense. This uses a general technical result, lemma 3.3, that will be appliedat several points. It allows to extract regularity results and estimates for

individual components of a 1-form from weak equations where boundaryconditions are imposed on the test 1-forms. In a subsection we discuss more

closely the space of weakly flat connections on a Riemann surface, *4°'P(E),and its quotients by the gauge group and the based gauge group.

Chapter 4 describes the crucial properties of the Lagrangian submanifolds

£ C A°'p(Tj) for which the boundary value problem (1.2) will be considered.

In particular, corollary 4.2 shows that the space of I/F1,p-connections with La¬

grangian boundary conditions is the closure of a space of smooth connections.

16 CHAPTER 1. INTRODUCTION

Moreover, a subsection introduces the main example £y of such Lagrangian

submanifolds, the set of connections on the boundary dY of a handlebodythat extend to a flat connection on Y.

Chapter 5 deals with Cauchy-Riemann equations with totally real bound¬

ary conditions for functions with values in a complex Banach space. We

establish the usual regularity results and estimates under one crucial as¬

sumption. In order to obtain ^-regularity results or estimates, the totallyreal submanifold (and hence also the complex Banach space) has to be mod¬

elled on a closed subspace of an Lp-space. This is the general setting for

the key part of the boundary value problem (1.2) and allows to deal with

the nonlocal Lagrangian boundary condition. The results in this chapter are

central for the proofs of theorems A, B, and C.

In chapter 6, we prove the regularity and compactness result for (1.2),theorems A and B, and chapter 7 establishes the Fredholm theory in the

compact case, theorem C. Finally, in chapter 8, we give an outlook on the

bubbling analysis at the boundary and prove some partial results.

The appendix gives a more detailed introduction into the basic notations

and constructions in gauge theory. This appendix is taken from [We], a

forthcoming monograph. The latter is an extensive exposition of the general

analytic background of gauge theory. In particular, it contains detailed proofsof Uhlenbeck's compactness and removable singularity theorems as well as

the regularity theorem for Yang-Mills connections. The generalizations to

manifolds with boundary, noncompact manifolds, and varying metrics seem

to not have been written up before. Moreover, this exposition has been set up

in such a way as to provide a suitable analytic framework for the treatment

of the nonlocal boundary conditions in the present thesis. So in a number

of places we will quote results from [We] without proof - these are generallywellknown but cannot be found elsewhere in the precise formulation that we

need.

Chapter 2

Floer homology and the

Atiyah-Floer conjecture

The Atiyah-Floer conjecture belongs into the general realm of interaction be¬

tween symplectic geometry and low dimensional topology. Important progress

in these areas has been made in the last twenty years starting with the work

of Donaldson [D2] on smooth four-manifolds, which was based on anti-self-

dual instantons, and with the work of Gromov [Gr] on pseudoholomorphic

curves in symplectic manifolds. In both subjects Floer, inspired by Conleyand Witten, introduced in the late eighties his idea of a Morse theory for

functionals on infinite dimensional spaces, where the critical points have in¬

finite index and coindex, but the relative indices are finite.

As in Morse theory, the chain complex is generated by the critical pointsof the functional, and the boundary operator is constructed by counting con¬

necting orbits between critical points. Floer's connecting orbits are finite

energy solutions of certain nonlinear elliptic equations arising from the gra¬

dient flow equation of the functional, which itself is not well-posed. Floer

carried out this program in two cases, [Fl, F2], which have been developedfurther by a large number of authors.

The instanton Floer homology HF'°st(F) of a 3-manifold Y is defined in

terms of the Chern-Simons functional on the space A(Y) of connections over

Y (on the trivial SU(2)- or a nontrivial SO(3)-bundle),

CS(A) = lj^(AAdA) + l([AAA]AA)). (2.1)

In this case the critical points are gauge equivalence classes (using only the

17

18 CHAPTER 2. ATIYAH-FLOER CONJECTURE

identity component of the gauge group) of flat connections, and the connect¬

ing orbits are the anti-self-dual instantons on 1 x 7 with finite Yang-Mills

energy.

In the symplectic case Floer considered two Lagrangian submanifolds L0

and L\ in a symplectic manifold (M, uj). The critical points of the symplecticaction functional on the space of paths connecting LQ to L\ are the constant

paths, that is the intersection points of L0 and L\. The connecting orbits are

pseudoholomorphic strips u : Mx [0,1] —) M of finite energy \ J |Vw|2, whose

boundary arcs s >-> u(s, 0) and s >-> u(s, 1) lie in L0 and L\ respectively. Here

one has to choose an almost complex structure J G End(TM), J2 = — t that

is compatible with the symplectic form, i.e. u(-, «/•) is a metric on M. Then

a pseudoholomorphic curve is a solution of

dsu + J(u) dtu = 0.

Under certain monotonicity assumptions this gives rise to Floer homology

groups HFfmp(M, L0, L\). Similarly, one can define Floer homology groups

HFfmp(M, ip) for symplectomorphisms <p G Diff(M,a>), [F3]. Here the crit¬

ical points are the fixed-points of <p, and the connecting orbits are finite

energy pseudoholomorphic curves u : M2 —> M satisfying a twist condition

associated with (p.

Atiyah and Floer conjectured the existence of natural isomorphisms be¬

tween the instanton Floer homology of 3-manifolds and the symplectic Floer

homology of Lagrangians or symplectomorphisms on moduli spaces of flat

connections associated with the 3-manifold. The starting point for these

conjectures is the Atiyah-Bott [AB, §9] picture of the moduli space of flat

connections over a Riemann surface as a symplectic quotient.

The moduli space of flat connections over a Riemann

surface

Let P —> E be a G-bundle over a Riemann surface E with a compact Lie group

G. The space of connections on P is an affine space A(P) = A + QX(E; gP)for any fixed reference connection A. It carries a symplectic structure: For

a,ßeTAA(P) = n1(^5p)

u)(a,ß) := J (aAß). (2.2)

19

The action of the gauge group G(P) on A(P) can be viewed as Hamiltonian

action of an infinite dimensional Lie group. The Lie algebra of G(P) is

£7°(E; 0p) and the infinitesimal action of £ G f2°(E; gP) is given by the vector

field

A(P) —>• ^(£;0p) =TAA(P)**• A ^ dA^.

This is the Hamiltonian vector field of the function A \-> ( (J,(A), £ ) on

A(P). Here (C£) = /e(C)^) denotes the inner product on the Lie algebra

S7°(E;0p) and // is the moment map of the gauge action,

A(P) _^ ^(E;0P)^'

A i—>• *FA.

Indeed, one has for all ß G ü1(E;0p)

w(^(A),/?) = J(dA^Aß) = -(*dAß,£),

where *d^ is the differential of the moment map at A G A(P). Now at an

irreducible 1 flat connection A G «4flat(-P) one nas an exact sequence

0 —> UeG{P) —> TAA —> LieG(P) —> 0

0 — n°(Z;gP) ^ Q'i^gp) ^ ^°(E;0p) — 0

Here the second arrow is the infinitesimal action at A and the third arrow

is the differential of the moment map at A. This shows that the quotient

/i_1(0)/£/(-P) = Aüa,t{P)/G{P) =' MP has the structure of a smooth manifold

near the equivalence class of A. (This can be made precise in the setting of

Banach manifolds - using the Sobolev completions of the space of connections

and the gauge group.) So the moduli space of connections MP is a smooth

manifold with singularities at the reducible connections. It can moreover

be seen as the symplectic quotient of the Hamiltonian gauge action on the

symplectic space of connections,

MP = A{P)//G(P) = ir\Q)/G{P).

So MP is a (singular) symplectic manifold with the symplectic structure

induced by (2.2).

1A connection A G «4flat(.P) is called irreducible if its isotropy subgroup of G{P) (thegroup of gauge transformations that leave A fixed) is discrete, i.e. d^lno is injective. For

a closed Riemann surface this is equivalent to d^|ni being surjective.

20 CHAPTER 2. ATIYAH-FLOER CONJECTURE

The Atiyah-Floer conjecture for mapping tori

A first version of the Atiyah-Floer conjecture for mapping tori was confirmed

by Dostoglou and Salamon [DS2]: Consider a nontrivial SO(3)-bundle P —> E

over a Riemann surface E with an orientation preserving automorphism

/ : P —> P. Similarly as above, the moduli space MP := Ana,t(P)/Go{P)of flat connections on P modulo the identity component Go {P) of the gauge

group is a compact symplectic manifold, and the pullback of connections

under / obviously induces a symplectomorphism cpf : MP —> MP. So one

has a symplectic Floer homology HF®ymp(</?/)• On the other hand, the au¬

tomorphism / induces an orientation preserving diffeomorphism / : E —y E

(by abuse of notation), which gives rise to an oriented closed 3-manifold, the

mapping cylinder Fj := I x E/ ~ with (t + 1, z) ~ (t, f(z)) for all i G R and

z G E. The same mapping cylinder construction yields a nontrivial SO (3)-bundle Pf over Yf. This can be used to define the instanton Floer homology

mr*(Yf).

Theorem 2.1 (Dostoglou,Salamon) Let f : P —> P be an orientation

preserving automorphism of a nontrivial SO (3)-bundle P —> E over a Rie¬

mann surface E. There is a natural isomorphism

HFrt(l/)-HFfmP(^/).

The proof relates pseudoholomorphic sections of a bundle over a cylinderwith fibre Mp to anti-self-dual instantons on the corresponding 4-manifold

(in which the fibre is replaced by E itself). The relation is established by an

adiabatic limit argument in which the metric on E converges to zero. More

precisely, a flow line of the symplectic Floer homology is a pseudoholomorphic

cylinder, u : M2 —) MP with the twist condition u(s,t + 1) = <pf(u(s,t)) and

dsu + J(u)dtu = 0.

Here the almost complex structure J on MP is induced by the Hodge operator

on E. One can lift u to a map A : R2 ->• A&a(P) with A(s,t + 1) = f*A(s,t).Then the pseudoholomorphic equation is equivalent to

dsA + *dtA = dA$ + *dA^,

where $, \t : M2 —> Q°(E, gP) are uniquely determined by A and satisfy the

same twist condition as A. Now one can view A + $ds + ^dt as connection

21

on M x Pf which satisfies

dsA - dA$ + *dtA - *dA^ = 0,

*FA = 0. (2.3)

On the other hand, a flow line of the instanton Floer homology is an (equiv¬alence class of an) anti-self-dual instanton on M x Py, i.e. a connection

A + $ds + *dt that satisfies

dsA - dA$ + *dtA - *dA^ = 0,

dt<S> - ds$> - [$, *] - e~2 * FA = 0. (2.4)

Here one can fix any e > 0 by rescaling the metric on E by the factor e2,since the instanton Floer homology is independent of the metric on the base

manifold. An adaptation of Uhlenbeck's compactness theorems shows that

sequences of such anti-self-dual instantons for e —> 0 converge (modulo gauge)to solutions of (2.3). Now the gauge equivalence classes of these solutions are

exactly the pseudoholomorphic cylinders. Conversely, an implicit function

argument shows that for sufficiently small e > 0 near every solution of (2.3)one finds a solution of (2.4). So this gives a bijection between the flow lines

of the symplectic and the instanton Floer homology. Moreover, the critical

points of both Floer homologies can be naturally identified as follows. A

connection A + ^dt G A{R x P) is flat if FA = 0 and À - dA^ = 0. So if one

makes \& vanish by a gauge transformation u G G(R x P) then one obtain

a constant path u*(A + ^dt) = A0 G Aüat{P)- Now such a constant path

comes from a flat conection A+ ^dt on Pf = Rx Pj ~ precisely when f*A0 is

equivalent by a gauge transformation in the identity component of the gauge

group to A0, that is the gauge equivalence class of A0 is a fixed-point of the

map 4>f induced by /* on the moduli space MP. This indentifies the critical

points of the instanton Floer homology with those of the symplectic Floer

homology.

The Atiyah-Floer conjecture for homology 3-spheres

The original version in [A2] of the Atiyah-Floer conjecture for Heegard split¬

tings of homology 3-spheres is still open: A Heegard splitting Y = Y0 Us Y\

of a homology 3-sphere is a decomposition into two handlebodies Y0, Y\ with

common boundary E. It gives rise to two Floer homologies. The moduli space

22 CHAPTER 2. ATIYAH-FLOER CONJECTURE

Me of flat connections on the trivial SU(2)-bundle over E is a finite dimen¬

sional symplectic manifold (with singularities) and the moduli spaces LYi of

flat connections over E that extend to Yt are Lagrangian submanifolds of Ms

(see chapter 4). Atiyah and Floer conjectured that the resulting LagrangianFloer homology should be isomorphic to the instanton Floer homology of Y.

Conjecture 2.2 (Atiyah,Floer) Let Y = Y0 Ue Y\ be a Heegard splitting

of a homology 3-sphere. Then there exists a natural isomorphism

HFmst(y) ^ HFfmP(ME,Lyo,JLYl).

Taubes [T] proved that the Euler characteristics agree. Salamon [Sal]outlined a program for a proof of the Atiyah-Floer conjecture for homology

3-spheres. The central point of this program is a Lagrangian boundary value

problem for anti-self-dual instantons that is motivated by the Chern-Simons

functional.

Chern-Simons functional on 3-manifolds with boundary

In this subsection we give a geometric motivation for the boundary value

problem that is treated in this thesis. The map

FA'- ct^ / (FAAa)

defines a 1-form T on the space of connections A(Y) on the trivial G-bundle

over a compact 3-manifold Y. Near a connection A G A(Y) with trivial

isotropy group the space of connections A(Y) can be seen as C/(y)-bundleover MY := A(Y)/G(Y). The 1-form T on this bundle is gauge invariant,and if Y is closed, then it is also horizontal, i.e. it vanishes on the fibres: For

all vertical tangent vectors a = d^Ç with £ G Q°(Y; g) Stokes' theorem gives

rA{aA0 = - f (dAFA,0+ f (FA,Ç) = 0. (2.5)Jy JdY

So for closed manifolds Y this 1-form descends to a 1-form on the (singu¬lar) moduli space MY, and moreover T is closed. Indeed, one can view

a,ß G fi1 (F; g) = TaA(Y) as (constant) vector fields on A(Y), then their

23

Lie bracket vanishes and one obtains by Stokes' theorem and due to dY = 0

dF(a,ß) = Va{F{ß))-Vß{F{a))

= / (dAaAß)- / {dAßAa)

= J (a Aß) = 0. (2.6)JdY

In fact, for closed Y the 1-form T is even exact - it is the differential of the

Chern-Simons functional (2.1). If Y has nonempty boundary dY = E then

the differential (2.6) does not vanish but is equal to the standard symplecticstructure u> defined in (2.2) on „4(E). To render T closed, it is natural to

pick a Lagrangian submanifold C C «4(E) and restrict T to the space

A{Y,C):={AeA(Y)\A\j,eC}.

(If £ C «4(E) is any submanifold, then the closedness of T is equivalent to

u\c = 0, and the maximal such submanifolds are precisely the Lagrangian

submanifolds.) In order that T again descends to a 1-form on the moduli

space A(Y,£)/G(Y) one has to assume that £ is gauge invariant and that

£ C «4flat(E) lies in the space of flat connections. 2 The first assumption

ensures that G{Y) acts on A(Y, £) and the second assumption renders T

horizontal, cf. (2.5). Under these assumptions £ descends to a (singular)Lagrangian submanifold in the (singular) moduli space of flat connections,

L := £/g(Z) C Me := Aat(£)/£(£).

In order to obtain a well defined Floer homology for such Lagrangians we shall

moreover assume that L is simply connected. (This ensures a monotonicity

property that gives control on the energies of flow lines between fixed critical

points.) Now in general, C is not simply connected, but its fundamental

group cancels with that of £/(E). This is the reason why T is not exact

but can only be written as the differential of the multi-valued Chern-Simons

functional

CSC(A) = \ j {{AAdA) + \{[AAA]AA))+ J J (A0(t) A À0(t)) dt.

2These two assumptions are equivalent if G is connected, simply connected, and has a

discrete center - as for example G = SU(2).

24 CHAPTER 2. ATIYAH-FLOER CONJECTURE

Here one has to choose a path A0(t) G £ with A0(0) = 0 (or any other fixed

reference connection in £) and A0(l) = ^|s- Unless £ is simply connected the

homotopy class of this path is not unique and hence the right-hand side is onlywell defined up to some positive integer. From now on we take G = SU(2),then this defines a functional CSc ' A(Y, £) —> M/47r2Z. A negative gradientflow line of this functional is a path A : R —y A(Y) satisfying

dsA + *FA = 0, A(s)|sG£ VsGR.

Equivalently, one can view this path as connection A = A + $ds G A(R x Y)in the special gauge $ = 0. Then the above equation is the anti-self-duality

equation for A. So the gauge equivalence classes of gradient flow lines of the

Chern-Simons functional are in one-to-one correspondence with the gauge

equivalence classes of solutions of the following boundary value problem for

connections Ä G «4(R x Y)

FA + *FÀ = 0,A„

A

(2.7)A|{s}xsG£ VsGM.

This is precisely the boundary value problem that we study in this thesis.

Floer homology for 3-manifolds with boundary

Fukaya set up a program to define Floer homology groups for 3-manifolds

Y with boundary dY = E using Lagrangian boundary conditions in the

moduli space Ms of flat connections, [Ful, Fu2]. For the definition of Floer

connecting orbits he uses a degeneration of the metric on Y to couple the

anti-self-duality equation for instantons on the interior of M x Y to the pseu¬

doholomorphic equation for strips in Ms with a Lagrangian boundary con¬

dition. In order that the moduli space Ms is a smooth symplectic manifold,

Fukaya uses SO(3)-bundles over Y that are nontrivial over the boundary dY.

The above discussion of the Chern-Simons functional suggests an alter¬

native approach by Salamon [Sal] that allows to use trivial SU(2)-bundles.This approach uses the solutions of (2.7) to define the Floer homology groups

HF1°st(y, L) for a 3-manifold with boundary dY = E and a Lagrangian sub¬

manifold L = £/0(E) C Ms. This is the starting point of his program for

the proof of the Atiyah-Floer conjecture 2.2. Fukaya's approach would also

fit into this program, however, this would require to extend his definition of

25

the Floer homology groups HF'"st(F, L) to trivial bundles and thus singularmoduli spaces. This thesis follows Salamon's approach and sets up the basic

analysis for the boundary value problem (2.7) that is required for the defini¬

tion of Floer homology groups.

The Floer complex will be generated by the critical points,

CF(y,L):= 0 Z([A]).

[A\£Al^(Y,C)/Ç{Y)

Here A^at(Y,£) denotes the set of irreducible flat connections A G Ana,t(Y)with Lagrangian boundary conditions A\s G £. 3 For any two such con¬

nections A+, A~ one then has to study the moduli space of Floer connecting

orbits,

M(A~, A+) = {Äe A(R xY)\Ä satisfies (2.7), lim Ä = ^j/^R x Y).

The analysis of the asymptotic behaviour of solutions of (2.7) should show

that the convergence at infinity is equivalent to the Yang-Mills energy of

the instanton being finite, hence these moduli spaces consist of connected

components of

M(C) = {Äe A{M x Y) | (2.7), yM{Ä) < oo}/£(K x Y).

Firstly, theorem A shows that the spaces of smooth connections and gauge

transformations in the definition of these moduli spaces can be replaced bysuitable Sobolev completions. Now AA(£) can be identified with the zeros

of a section of a Banach space bundle. Theorem C then is one step towards

showing that this moduli space is a smooth manifold. It asserts that in the

compact model case, the linearizations of the section at its zeros are Fred¬

holm operators. The analysis of the asymptotic behaviour of the solutions

should be combined with this Fredholm theory (and a perturbation of the

equations (2.7) ) to show that A4(A~, A+) is a disjoint union of smooth mani¬

folds of the dimension /J>(A~) — n(A+) + 8Z. (These connected components

are distinguished by the homotopy class of the path in A(Y, £) running from

3 There should be no reducible flat connections with Lagrangian boundary conditions

other than the gauge orbit of the trivial connection. This will be guaranteed by certain

conditions on Y and L, for example this is the case when L = Ly for a handlebody Y'

with dY' = S such that Y ris Y' is a homology-3-sphere.

26 CHAPTER 2. ATIYAH-FLOER CONJECTURE

A' to A+.) Here fj, : A\&i(Y,C) ->• Z/8Z is a (mod 8)-grading on the Floer

complex. Now a monotonicity property provides a fixed Yang-Mills energy

for the connections in the fc-dimensional part Aik(A~, A+) of the space of

connecting orbits.

Theorem B is one step towards a compactification of these moduli spaces

Aik(A~, A+). It proves their compactness under the assumption of an In¬

bound on the curvature for p > 2, whereas the Yang-Mills energy is onlythe L2-norm. So an analysis of the possible bubbling phenomena should

then lead to the required compactification. In the case of index difference

fjt(A-) - ß{A+) = 1 (mod 8), this will show that M1{A-,A+)/R is a finite

set (after dividing out the time-shifts). Thus the Floer boundary operator

can be defined by

d{[A-]):= £ #(^1(^-,^+)/M)([A+]).ß(A+)^(A-)-l

As usual in Floer homologies, dod = 0 should then follow from a glueing the¬

orem that identifies the broken flow lines f][A] M1(A~,A)/R x Ai1{A, A+)/Rwith the boundary of the compactification of Ai2(A~, A+)fR. (The number

of broken flow lines from A~ via some A G A^&t (Y, £) to A+ gives the factor

in front of ( [A+] ) in dd( [A~] ). This number is even, so it vanishes in Z2,since it is the number of boundary points of a compact 1-manifold. If counted

with signs according to certain choices of orientations, then it also vanishes in

Z.) One then obtains the Floer homology groups HF^F, L) := H*(CF, d).In order to prove that these are independent of the metric on Y, one

defines a chain homomorphism between the Floer complexes correspondingto different metrics g~ and g+ on Y. This is done completely analogousto the definition of d by counting the solutions of (2.7) for a metric on

M x F that interpolates between g~ and g+ (see example 1.4). Finally,

one has to prove that the thus defined isomorphism of Floer homologies is

independent of the chosen interpolating metric, i.e. find a chain homotopy

equivalence between the chain homomorphisms of two different interpolatingmetrics. This homomorphism of the Floer complexes is again defined by

counting elements of a moduli space. This time however, one chooses a pathbetween the two interpolating metrics, then the moduli space contains pairsof metrics in this path and solutions of (2.7) with respect to this metric. The

compactification of these moduli spaces is the reason for considering varyingmetrics in theorem B.

27

The program for a proof of the Atiyah-Floer conjectureand the significance of this thesis

The first step of the program by Salamon [Sal] for the proof of the Atiyah-Floer conjecture is to define the instanton Floer homology HFst(Y, L) for a

3-manifold with boundary dY = E and a Lagrangian submanifold L C Ms

in the moduli space of flat connections. The basic analytic framework for

this Floer theory is set up in this thesis. The regularity theorem A shows

that the boundary value problem whose solutions will be the connecting or¬

bits is wellposed. The compactness theorem B is a major step towards the

compactification of the moduli spaces of connecting orbits. It remains to

analyse the bubbling phenomena at the boundary - these remaining obsta¬

cles for the compactification of the moduli spaces are discussed in chapter 8.

Theorem C sets up the Fredholm theory (leading to the smoothness of the

moduli spaces) for the compact model case. Here it remains to analyse the

asymptotic behaviour of finite energy solutions of the boundary value prob¬lem (2.7) - which should be a straightforward analogon of the closed case -

and combine this with theorem C to a Fredholm theory in the noncompact

case. Suitable perturbations of the anti-self-duality equation as in the closed

case then should lead to the required transversality result giving the mod¬

uli spaces the structure of finite dimensional smooth manifolds. Togetherwith the compactification of the moduli spaces this allows to define the Floer

boundary operator d. Finally, the proof of d o d = 0 and the metric indepen¬dence of the resulting homology require several glueing theorems that should

work analogously to Floer's original glueing theorem [Fl].

The next step in the program for the proof of the Atiyah-Floer conjectureis to consider a Heegard splitting Y = Y0 Ue Yi of a homology 3-sphere, and

replace the instanton Floer homology of Y by the Floer homology of the

3-manifold [0,1] x E with boundary Ë U E and the Lagrangian submanifold

LYo x LYl C MEuS in the moduli space of flat connections over Ë U E.

Here dY0 = E and dYx = Ë is the same Riemann surface with oppositeorientation. Let £Yi be the space of flat connections over E that extend to

flat connections over Y{. Then £ := £Yo x £Yl C .4(E) x .4(E) = i(ËflS) is

a Lagrangian submanifold as considered in (2.7) and £/Ç/(ËnE) = LYo x LYl

withLy, =£YJG(Z).In [Sal] it is conjectured that these two Floer homologies should be iso¬

morphic.

28 CHAPTER 2. ATIYAH-FLOER CONJECTURE

Conjecture 2.3 If Y = Y0 Us Yi is a homology 3-sphere then there is a

natural isomorphism

HFmst(y) ^ HF^QO, 1] x E,Lyo x LYl).

Firstly, the Floer complexes can be identified as follows. Consider a

flat connection A G Aûg,t(Y). Thicken E C Y and slightly shrink the

Yt such that one obtains a disjoint union Y = YQ n [0,1] x E n Y\_. Now

-4|[o,i]x£ G Aat([0,1] x E) with the boundary values A\{0}xS = A\dYo G £Yoand ^4|{i}xs = A\gYl G £Yl. So the main task is to identify the connectingorbits.

The idea of the proof is to choose an embedding (0,1) x E <—>• Y startingfrom a tubular neighbourhood of E C Y at t = \ and shrinking {£} x E to the

1-skeleton of Yt for t = 0,1. Then the anti-self-dual instantons on M x Y pullback to anti-self-dual instantons on M x [0,1] x E with a degenerate metric

for t = 0 and t = 1. On the other hand, one can consider anti-self-dual

instantons on M x [e, 1 — e] x E with boundary values in £Yo and £Yl. As

e —y 0, one should be able to pass from this genuine boundary value problemto solutions on the closed manifold Y.

The final part of the proof of the Atiyah-Floer conjecture would be to

establish the following conjecture [Sal].

Conjecture 2.4 If Y = Y0 Us ^i is a homology 3-sphere then there is a

natural isomorphism

HF^UO, 1] x E, LYo xLYl)^ HFs/mP(Ms, LYo, LYl).

Again, one sees easily that the Floer complexes (generated by the critical

points) are naturally isomorphic as follows. In the instanton Floer homologythe critical points are the equivalence classes of flat connections on [0,1] x E

with boundary values in £Yo and £Yl at t = 0 and t = 1 respectively. Theycan be written as A + *dt with A(t) G f^(E;0) and *(t) G ^°(E;0) for

all t G [0,1]. One can make ^ vanish by a gauge transformation, then

the flatness condition A — d^^ = 0 implies that A is t-independent, so

A(0) = A(l) G £Yo n £Yl. Thus the critical points here can be identified

with intersection points of the Lagrangian submanifolds LYo and LYl in the

moduli space Ms - which are exactly the critical points of the symplecticFloer homology.

29

So the proof of this conjecture requires an adaptation of the adiabatic

limit argument in [DS2] to boundary value problems for anti-self-dual in¬

stantons and pseudoholomorphic curves respectively in order to identify the

moduli spaces of connecting orbits. Here one again deals with the boundaryvalue problem (2.7) studied in this thesis. As the metric on E converges

to zero, the solutions, i.e. anti-self-dual instantons on R x [0,1] x E with

Lagrangian boundary conditions in £Yo,£Yl C Ms should be in one-to-one

correspondence with connections on R x [0,1] x E that descend to pseudo¬

holomorphic strips in Ms with boundary values in LYo and LYl.

30 CHAPTER 2. ATIYAH-FLOER CONJECTURE

Chapter 3

Weakly flat connections

In this chapter we consider the trivial G-bundle over a closed manifold E

of dimension n > 2, where G is a compact Lie group. Fix p > n. Then a

connection A G A0,P(T,) is called weakly flat if

f (A,d*u-l{-l)n*[AA*u]) = 0 \/ueQ2{Z,g). (3.1)

For sufficiently regular connections, (3.1) is equivalent to the connection be¬

ing flat, that is FA = dA + ^[A A A] = 0.We denote the space of weakly

flat L^-connections over E by

„4°ft(E) := {A G -4°'P(E) | A satisfies (3.1)}.

One can check that this space is invariant under the action of the gauge

group Gl,p{^)- Here p > 2 is required in order that (3.1) is welldefined, and

Gl,p(E) is welldefined for p > n, see e.g. [We, Appendix B].The next theorem shows that the quotient *4fl^t(E)/'Q1,%'(E) can be identi¬

fied with the usual moduli space of flat connections *4flat(E)/Ç/(E) - smooth

flat connections modulo smooth gauge transformations.

Theorem 3.1 For every weakly flat connection A G ^lfl'^t(E) there exists a

gauge transformation u G G1,p{^) such that u*A G *4flat(E) is smooth.

The proof will be based on the following L^-version of the local slice

theorem, a proof of which can be found in [We, Theorem 9.3].

31

32 CHAPTER 3. WEAKLY FLAT CONNECTIONS

Proposition 3.2 Fix a reference connection A G *4°'P(E). Then there exists

a constant 5 > 0 such that for every A G A0,P(T,) with \\A — A\\p < 5 there

exists a gauge transformation u G Gl'p{^) such that

J(u*A_Â1dÂV) = 0 V?7GCoo(E,0). (3.2)

Equivalently, one has for v = u^1 G Gl,p{^)

j {v*A - A , dAr,) = 0 \/V G C°°(E, g).

The weak flatness together with the weak Coulomb gauge condition (3.2)form an elliptic system, so theorem 3.1 then is a consequence of the followinglemma. For closed manifolds M = E, this regularity result is essentiallydue to the Hodge decomposition of i7-regular 1-forms. In the case when

M = E is in fact a Riemann surface, this result directly follows from the

regularity theory for the weak Laplace equation. However, the lemma also

holds on manifolds with boundary, and it yields componentwise regularityresults. This will be useful for the proof of theorem C in chapter 7. Here we

use the following notation:

C|°(M) = {(/»GCOO(M)|0|9M = O},

C(M) = {^GC°°(M)|f|9M = 0}.

Lemma 3.3 Let (M,g) be a compact Riemannian manifold (possibly with

boundary), let k G N0 and 1 < p < oo. Let X G T(TM) be a smooth vector

field that is either perpendicular to the boundary, i.e. X\gM = h v for some

h G C°°((9M); or tangential, i.e. X\gM G T(TdM). In the first case let

T = Cg°(M), m the latter case let T = C^(M). Then there exists a constant

C such that the following holds:

Let f G Wk'p(M), 7 G Wk'p(M,A2T*M), and suppose that the 1-forma G Wk'p{M,T*M) satisfies

f {a,dr])= f f-f} yVeC^(M)7Jm Jm

f (a,d*co)= [ (j,u) Vu = d{<j) LXg), cj>eT.Jm Jm

Then a(X) G Wk+1'P(M) and

\\a(X)\\Wk+i,P < Cdl/H^/fc,? + ||7||wfc>p + IMIty^pJ-

33

Remark 3.4 In the case fc = 01et--|--V = l, then the weak equations for a

can be replaced by the following: There exist constants c\ and c2 such that

[ (a,dV) <Cl||77||p. VveC^iM),Jm

/ (a, d*d((ß- txg)) < c2 IMI^lp*). ^4>&T.Jm

The estimate then becomes ||û!(X)||^i,p < C{c\ + c2 + ||o;||p) .

The regularity and estimates claimed here will follow from weak Laplace

equations. We abbreviate A := d*d. Then the proof will use the followingstandard elliptic theory for the Laplace operator with Dirichlet or (inho-mogenous) Neumann boundary conditions, which can for example be found

in [GT] and [We, Theorems 2.3',3.2,D.2].

Proposition 3.5 Let k G N; then there exists a constant C such that the

following holds. Let f G Wk~1'p(M) and G G Wk'p(M) and suppose that

u G Wk'p(M) is a weak solution of the Dirichlet problem (or the Neumann

problem with inhomogenous boundary conditions), that is for all iß G C^°(M)(or for all iß e C(M))

u AißM

f-iß+ f G-iß.M JdM

Then u G Wk+l*{M) and

\\u\\Wk + l,v < C(j\f\\Wk-l,p + HGHjyfe.p + HttHtyfe.p).

In the special case k = 0 there exists a constant C such that the followingholds: Suppose that u G If (M) and that there exists a constant c such that

for all iß G Q°(M) (or for all iß G C?{M))

JJM

U a.lß < C i,p*w1*

Then u G Wlj>(M) and

\u\\wi,P < C(c-\- \\u\\Lp).

34 CHAPTER 3. WEAKLY FLAT CONNECTIONS

Proof of lemma 3.3 and remark 3.4 :

Let a" G C°°(M, T*M) be an L^-approximating sequence for a such that

au = 0 near dM. Then one obtains for all (f> G T

f a(X)-A<f>= lim ( f {£xav, d</>)- / ( ixdau, <ty)Jm "-*30 \Jm Jm

= lim f - / ( a" , £xd</> ) - / ( a" ,divX • d<ß )

v^°° \ JM JM

~ \ {o?, Ly^Cxg) - \ {dav ,ixgAd<ß)Jm Jm

= / (a, d(-£xcf) - divX <j>)) - I ( a, d*(txg A d0) )Jm Jm

+ / {<*,</> d(divA") - Ly^Cxg)JM

= [ (f,-£x</>-divX <!>) + [ (<y,d{<f>-ixg))JM JM

- I (a,d*{(ß-dcxg))+ \ ( a , <j> d(divX) - tYd^£xg).JM JM

Here the vector field Yâ(p is given by tyd(/)g = d(ß. In the case k > 1 further

partial integration yields for all <f> G T

f a(X)-Acß = f F-</> + f G-<f>,Jm Jm Jom

where F G Wk-1'P(M), G G Wk'p(M), and for some constant C

\\F\\Wk-i,P + ||G||^fc,p < C(||/||v^fc,p + UtIIvf^.p + ||a;||^fc,p).So the regularity proposition 3.5 for the weak Laplace equation with either

Neumann (T = C£°(M)) or Dirichlet (T = Cf){M)) boundary conditions

proves that a(X) G Wk+1'P(M) with the according estimate.

then there is a constant C such that for all <f> G T

In the case k = 0 one works with the following inequality: Let -V + - = 1,

/J M

a(X) A(ß <C(ll/llp+ll7llp+l|a||P)ll0llw^'

(Under the assumptions of remark 3.4, one simply replaces ||/||p and ||7||pby c\ and c2 respectively.) The regularity and estimate for a(X) then follow

from proposition 3.5.

35

Proof of theorem 3.1 :

Consider a weakly flat connection A G Aq^ÇE). Let S > 0 be the constant

from proposition 3.2 for the reference connection A and choose a smooth

connection À G «4(E) such that \\A — A\\p < 5. Then by proposition 3.2

there exists a gauge transformation u G G1,P(E) such that

L>{u*A-Ä,dÄr1) = 0 Vr?GCoo(E,0).

Now lemma 3.3 asserts that a := u*A — À G LP(T,, T*E®0) is in fact smooth.

(By the definition of Sobolev spaces via coordinate charts it suffices to prove

the regularity and estimate for a(X), where X G T(TE) is any smooth vector

field on E.) This is due to the weak equations

f{a,dr}) = - /(*[«A*I],r7) Vr? G C°°(S,fl),Js Je

[{a, d*u) = - f (dÄ + ±[u*AAu*A],u) Vw6fi2(S,s).is is

Firstly, the inhomogeneous terms are of class L2, hence the lemma asserts

W1' 2 -regularity of a and u*A. Now if p < 2n, then the Sobolev embedding

gives LPl-regularity of u*A with pi := -^— (in case p = 2n one can choose

any pi > 2n). This is iterated to obtain 17^-regularity for the sequence

Pj+i = 2"Pj (or any pj+1 > 2n in case pj > 2n) with p0 = p. One checks

that pj+i > 9pj with 9 = ^- > 1 due to p > n. So after finitely many

steps this yields IT^-regularity for some q = 2j- > n. The same is the case

if p > 2n at the beginning. Next, if u*A is of class Wk,q for some k G N,then the inhomogeneous terms also are of class Wk,q and the lemma asserts

the T47fe+1'<?-regularity of a and hence u* A. Iterating this argument proves

the smoothness of u*A = A + a.

Weakly flat connections over a Riemann surface

Now we consider more closely the special case when E is a Riemann surface.

Theorem 3.1 shows that the injection .4flat(£)/£(£) ^ A°^t{T,)/G1'p(^) in

fact is a bijection. These moduli spaces are identified and denoted by Ms.

Furthermore, the holonomy induces a natural bijection from Ms to the space

of conjugacy classes of homomorphisms from 7Ti(E) to G, see theorem A.2

Ms := .4&E)/^(E) ^ Aat(£)M£) = Hom(7n(S),G)/~.

36 CHAPTER 3. WEAKLY FLAT CONNECTIONS

From this one sees that Ms is a finite dimensional singular manifold. If we

assume G = SU(2), then Ms has singularities at the product connection and

at the further reducible connections - corresponding to the connections for

which the holonomy group is not SU(2) but only {11} or is conjugate to the

maximal torus S1 C SU(2).xAway from these singularities, the dimension

of Ms is 6g — 6, where g is the genus of E. (The arguments in [DS1, §4] show

that T[,4]Ms — kerd^/imd^ = hA has dimension 3 • (2g — 2) at irreducible

connections A.)For the same reasons the space of weakly flat connections A^t (E) is in

general not a Banach submanifold of A°'P(T,) but a principal bundle over a

singular base manifold. To be more precise fix a point z G E and consider

the space of based gauge transformations, defined as

£*(E) := {u G ^(E) | u(z) = 11} .

This Lie group acts freely on „4fl^t(E). The quotient space ^lfl'^t(E)/^'p(E)can be identified with Hom(7Ti(E), G) via the holonomy based at z. This

based holonomy map pz : „4fl^t(E) —> Hom(7Ti(E), G) is defined by first

choosing a based gauge transformation that makes the connection smooth

and then computing the holonomy. Now pz gives >4fl^t(E) the structure of

a principal bundle with fibre C7]'P(E) over the finite dimensional singularmanifold Hom(7Ti(E), G),

^1'P(S)^^(E)^Hom(7r1(E),G).

Note that this discussion does not require the Riemann surface E to be

connected. Only when fixing a base point for the holonomy map and the

based gauge transformations one has to adapt the definition. Whenever

E = (J"=1 Ej has several connected components E,, then 'fixing a point z G E'

implicitly means that one fixes a point z% G E? in each connected component.

The group of based gauge transformations then becomes

£'p(Uli s0 := {u e £1>p(£) | <zA = 1 Vi = 1,..., n) .

1The holonomy group of a connection is given by the holonomies of all loops in S, cf.

appendix A. Now the isotropy subgroup of Q(S) of the connection is isomorphic to the

centralizer of the holonomy group, see [DK, Lemma 4.2.8].

Chapter 4

Lagrangians in the space of

connections

Consider the trivial G-bundle over a (possibly disconnected) Riemann surface

E of (total) genus g. Here again G is assumed to be a compact Lie group

and 0 denotes its finite dimensional Lie algebra. There is a gauge invariant

symplectic form to on the space of connections „4°'P(E) for p > 2 defined as

follows. For tangent vectors a, ß G LP(Y,, T*E®0) to the affine space „4°'P(E)

u{a,ß)= [ {a Aß). (4.1)

The action of the infinite dimensional gauge group Gl,p(^) on the symplecticBanach space (A0,p(T,), to) is Hamiltonian with moment map A \-y *FA (moreprecisely, the equivalent weak expression in (IT1,p*(E,0))*). So ME can be

viewed as the symplectic quotient A°'PÇE)//G1,PÇE) as was first observed by

Atiyah and Bott [AB], cf. chapter 2. However, 0 is not a regular value of

the moment map, so Ms is a singular symplectic manifold. Due to these

singularities at the reducible connections we prefer to work in the infinite

dimensional setting.A Lagrangian submanifold £ C *4°'P(E) is a Banach submanifold that

is isotropic, u>\tac = 0 for all A G £, and is of maximal dimension. In this

infinite dimensional setting, the latter condition can be phrased as

TAA°>P (E) = TA£ e J Ta£ VAe£.

Here one has to choose an ^-compatible complex structure J G End T„4°'P(E),i.e. such that J2 = —11 and u>(-, J-) defines a metric (The latter condition

37

38 CHAPTER 4. LAGRANGIANS

is in fact independent of the choice of J since the space of w-compatible

complex structures is connected [MSI, Proposition 2.48].)Now note that the Hodge * operator is an ^-compatible complex structure

since o>(-, *) is the L2-inner product. For all a, ß G LP(T,, T*E <g> 0)

u(a,*ß) = f{aA*ß) = (a,ß)L2. (4.2)Js

Hence a Banach submanifold £ C „4°'P(E) is Lagrangian if and only if it is

isotropic, u\c = 0, and totally real, i.e. for all A G £

LP(E,T*E®0) = TAC®*TAC. (4.3)

Suppose that £ is Lagrangian and in addition £ C *4fl^t(E), then one also

has the twisted Hodge decomposition (see e.g. [Wa, Theorem 6.8])

I7(E,T*E®0) = imdAeimd^e/4 (4.4)

where h\ = kerd^ D kerd^. The assumption £ C «4fl^t(E) also ensures that

£ is gauge invariant if G is connected and simply connected. On the other

hand, the gauge invariance of £ implies £ C *4fl'.ft(E) if the Lie bracket on G

is nondegenerate (i.e. the center of G is discrete). So for example in the case

G = SU(2) both conditions are equivalent.In general we will consider Lagrangian submanifolds £ C *4°'P(E) and

assume that they are both gauge invariant and contained in Aq^(T,). In that

case £ descends to a (singular) submanifold of the (singular) moduli space

of flat connections,

L := £/^(E) C<Pt(E)/^(E) =: ME.

This submanifold is obviously isotropic, i.e. the symplectic structure in¬

duced by (4.1) on Ms vanishes on L. Moreover, its tangent spaces have

half of the dimension of those of Ms, so L C Ms is a Lagrangian subman¬

ifold. Indeed, in the Hodge decomposition (4.4) imd^ is the complementof kerd^ = T^^Lj'^t(E), imd^ is the tangent space to the orbit of G1,p{^)through A, and so h\ = T^jMs- Now compare this with the decomposition

(4.3). Here imd^ C TA£ and imd^ C *TA£ are mapped to each other bythe complex structure *, and thus the intersection of hA with TA£ is of half

dimension - this is exactly T[a\£-

39

Moreover, our assumptions on the Lagrangian submanifold ensure that

the holonomy map pz : £ —y Hom(7Ti(E), G) based at z G E is welldefined

and invariant under the action of the based gauge group G\'p(^)- Note that

Hom(7Ti(E), G) naturally embeds into Hom(7Ti(E\{2;}), G), which is a smooth

manifold diffeomorphic to G2a. This gives Hom(7Ti(E), G) a differentiable

structure (that is in fact independent of z G E), however, it is a manifold

with singularities. In the following lemma we list some crucial propertiesof the Lagrangian submanifolds £ that we will deal with. Here we use the

notation

IT^(E,0) := {i G W^{H,q) I t(z) = 0}for the Lie algebra T1Gl'p{^) of the based gauge group. (If E is not connected

then as before one fixes a base point in each connected component and mod¬

ifies the definition of W^'p(Ti,g) accordingly.) Moreover, in this chapter, we

will denote the differential of a map 0 at a point x by Tx(ß in order to dis¬

tinguish it from the exterior differential on differential forms, d^, associated

with a connection A.

Lemma 4.1 Let £ C A0,P(T,) be a Lagrangian submanifold and fix z G E.

Suppose that £ C „4fl^t(E) and that £ is invariant under the action of Gl'p (E)Then the following holds:

(i) L := £/Gl'pÇE) ts a smooth manifold of dimension m = g dim G and

the holonomy induces a diffeomorphism pz : L —y M to a submanifold

McHom(7r1(E),G).

(%%) £ has the structure of a principal Ç?]'P(E) -bundle over M,

£*(£) ->• £ -^ M.

(in) Fix A G £. Then there exists a local section 4> : V —y £ over a neigh¬bourhood V C Rm of 0 such that </>(0) = A and pz o (f> is a diffeomor¬

phism to a neighbourhood of pz(A). This gives rise to Banach subman¬

ifold coordinates for £ C A0,P(T,), namely a smooth embedding

6 : W ^ -4°'P(E)

defined on a neighbourhood W C W^{E,g) x Rm x W}>p{^g) x Rm

of zero by

0(fo,«o,fi,vi) := exp(fo)Xuo) + *°U£i + *T0<K«i)-

40 CHAPTER 4. LAGRANGIANS

Moreover, if A is smooth, then the local section can be chosen such that

the image is smooth, (ß : V —y £ Pi .4(E). Now the same map O is a

diffeomorphism between neighbourhoods of zero in W^,p(Tj,g2) x R2m

and neighbourhoods of A in *4°'P(E) for all p > 2.

One reason for this detailed description of the Banach submanifold charts

is the following approximation result for Ly^-connections with Lagrangian

boundary values.

Corollary 4.2 Let £ C A0,P(T,) be as in lemma 4-1 and let

Q CH:= {(s,t) el2 | t > 0}

be a compact submanifold. Suppose that A G Al,p{Vt x E) satisfies the bound¬

ary condition

%o)xse/: V(s,0)GcKl (4.5)

Then there exists a sequence of smooth connections Av G Ai$1 x E) that

satisfy (4-5) and converge to A in the W1,p-norm.

Proof of corollary 4.2:

We decompose A = $ds + \&di + B into two functions $, * G W1,p(fi x E, 0)and a family of 1-forms B G Wl*{ïl x E, T*E®0) on E such that B(s, 0) E £

for all (s, 0) G dCl. Then it suffices to find an approximating sequence for B

with Lagrangian boundary conditions on a neighbourhood of Q D dM. This

can be patched together with any smooth iyliP-approximation of B on the

rest of fl and can be combined with standard approximations of the functions

$ and \& to obtain the required approximation of A.

So fix any (s0, 0) G Q D dM and use theorem 3.1 to find u0 G C/llP(E) such

that Aq := Mq_B(s0,0) is smooth. Lemma 4.1 (iii) gives a diffeomorphism6 : W ->• V between neighbourhoods W C W^p(E,g2) x R2m of zero and

V C _4°'P(E) of A0. This was constructed such that C°°(E,02) x R2rn is

mapped to .4(E) and such that 9:Wn W2'p{Z,g2) xl2ra^Vn ^(E)also is a diffeomorphism. Now note that B G C°(Q,A0,PÇE)). Hence there

exists a neighbourhood U C fl of (s0, 0) and one can choose a smooth gauge

transformation u G <?(E) that is W1,p-close to u0 such that u*B(s,t) G V

for all (s,t) G U. Now we define f = (&,&) : £/ - W>p(E,02) and

u = (v0,wi) : £/ - M2m by 0(C(s,t),v(s,t)) = u*B(s,t). Recall that B is of

class Wl'p on[/xS, hence it lies in both W1J"(U, A°'P{Z) and D>{U, Al*{Y).

41

Thus £ G W^p{U,W^p{^,g2)) n Lp(U, W2'P(E,S2)) and w G Wlj'(t/,M2m))and these satisfy the boundary conditions £1 |t=o = 0 and v\ |t=o = 0 due to the

Lagrangian boundary condition for B. Now there exist £v G C°°(U x S,02)and vv G C°°(f/, R2m) such that £y —)• £ and vv —y v in all these spaces,

£"(•,2) = 0, £i|t=o = 0, and Ui|t=o = 0. (These are constructed with

the help of mollifiers as in lemma 5.8; also see remark 5.9. One first re¬

flects £ at the boundary and mollifies it with respect to U to obtain ap¬

proximations in C°°(U, ITr2'p(E,02)) with zero boundary values. Next, one

mollifies on E, and finally one corrects the value at z.) It follows that

B"(s,t) := (u_1)*0(£(s, t), v(s, t)) is a sequence of smooth maps from U

to »4(E) which satisfies the Lagrangian boundary condition and converges to

B in the W1,p-norm.

Now fi n dM is compact, so it is covered by finitely many such neigh¬bourhoods Ui on which there exist smooth ly^-approximations of B with

Lagrangian boundary values. These can be patched together in a finite pro¬

cedure since the above construction allows to interpolate in the coordinates

between £", vv and other smooth approximations £', v' (arising from approxi¬mations of B on another neighbourhood U' in different coordinates) of £ and

v respectively. This gives the required approximation of B in a neighbour¬hood of Q n dM. and thus finishes the proof.

Proof of lemma 4.1:

Fix A G £ and consider the following two decompositions:

Lp(E,T*E®0)=TA£e*TA£ (4.6)

= dAW^(E,0) 8 *dAW^(E,0) 0 hA,

where Ha is a complement of the image of the following Fredholm operator:

W^p(E,g)xW^p(E,g) — i7(E,T*E(g)0)A'

(£,0 i— cU£ + *dAc

To see that Da is Fredholm note that for every A G >4fl^t(E) the operator

Da is injective and is a compact perturbation of Do- Hence moreover the

dimension of cokerl?^ (and thus of ha) is the same as that of cokerDo-

In the case A = 0 one can choose the space of 0-valued harmonic 1-forms

h1 = kerdnkerd* as complement ho. So ha must always have the dimension

dim ha = dim hl = 2g dim G = 2m. (Note that in general one can choose

ha to contain hA, but this might not exhaust the whole complement.)

42 CHAPTER 4. LAGRANGIANS

Due to the ^'p(S)-invariance of £ the splittings (4.6) now imply that

there exists an m-dimensional subspace LA C Ka such that

TA£ = dAW^p(^g)®LA.

So TA£ is isomorphic to the Banach space W^,p(T,,g) x Rm via d^ © F

for some isomorphism F : Rm —y LA. Here we have used the fact that d^

is injective when restricted to Wz1,p(E,fl). Now choose a coordinate chart

$ : Ta£ -^ £ defined near $(0) = A, then the following map is defined for

a sufficiently small neighbourhood V C Rm of 0,

.

#*(£) x V/ — £

(u,v) i—>• «*($o(To$)_1oF(r;)).

We will show that this is an embedding and a submersion (and thus a dif¬

feomorphism to its image). Firstly, T(10)^ : (£,iu) *-y d^£ + Fro is an

isomorphism. Next, note that \&(m, f) = m*^(11, w) and use this to calculate

for all u E £]'P(E), £ G 1T^P(E, 0), and v, w E Rm

T(„,„)* : (fu,tu) ^tT^d*^+ T(M)tf(0,tu))^

One sees that «(T^^VIJ)«-1 is a small perturbation of T^o)^, hence one can

choose V sufficiently small (independently of u) such that T(U)V)^ also is an

isomorphism for all v E V. So it remains to check that \& in fact is globally

injective.

Suppose that u,u' E Ç?]'P(E) and v,v' E V such that ^(u,v) = ty(u',v').Rewrite this as $(l,v) = ^f(ü,v') with ü := u'u'1 G £]'P(E). Now bythe choice of a sufficiently small V the norm 11^(11, v) — ty(l,v')\\p can be

made arbitrarily small. Then the identity ^(11, v) = ü*^(l, v') automatically

implies that ü is C°-close to 1. (Otherwise one would find a sequence of

^-connections A" -> A and vv E £]'P(E) such that \\uv*Av - Av\\p -> 0

but dco{u\t) > A > 0. However, from {uv)-lduv = vv*Av - {uvY1Avuvone obtains an Lp-bound on duv and thus finds a weakly VT^-convergentsubsequence of the uv. Its limit u E G\'p(^) would have to satisfy u*A = A,hence u = 11 in contradiction to dco(u, 11) > A > 0.) So one can write

ü = exp(£) where £ G W^'P{E, g) is small in the L°°-norm. Next, the identity

ü-ldü = *(1, v) - ü_1tf (J, v')ü

shows that ||£||wi.p will be small if V is small (and thus ü is C°-close to 11).Hence if V is sufficiently small, then (ü,v') and (11, v) automatically lie in a

neighbourhood of (11, 0) on which ^ is injective, and hence u = u' and v = v'.

43

We have thus shown that \& : C/]'P(E) x V —y £ is a diffeomorphism to its

image. This provides manifold charts iß : V —y £/Ç/]'p(E), v (->• [\&(1, w)] for

L := £/Gl'p(E)- Now fix 2g generators of the fundamental group 7Ti(E), then

the corresponding holonomy map pz : L —y G x • • • x G is an embedding,so its image M C Hom(7Ti(E), G) is a smooth submanifold. This proves (i).For (ii) the diffeomorphism $> gives a bundle chart over U := pz(iß(V)) C M,

namely* o (id x (pz o V)"1) : £'P(E) xW^£.

Furthermore, the local section for (iii) is given by (ß(v) := \&(1, f). However,this is a map ci) : V —y £; it does not necessarily take values in the smooth

connections. Now if A E £ fl »4(E) is smooth, then for a sufficiently small

neighbourhood V this section can be modified by gauge transformations such

that <ß : V —y £ fl .4(E). To see this, note that the gauge transformations in

the local slice theorem are given by an implicit function theorem: One solves

D{v,Ç) = 0 for £ = £(i>) G IT1'p(E,0) with the following operator:

VxW^i^g) — imd^ C (W^(E,g)y(v,Q ^ d'A(exp(0*(ß(v) - A) .

Here d^ denotes the dual operator of d^ on Wl,p* (E, 0). One has D(0, 0) = 0

and checks that <92-D(0, 0) : £ —y d'Ad^£ is a surjective map to imd^, see e.g.

[We, Lemma 9.5]. The implicit function theorem [L, XIV,Theorem 2.1] then

gives the required gauge transformations exp(£(v)) G £/llP(E) that bring (ß(v)into local Coulomb gauge and thus make it smooth. (By construction (ß(v) is

weakly flat, then see the proof of theorem 3.1.) This modification by gauge

transformations does not affect the topological direct sum decomposition

TA£ = dAW^p(Z,g)®imT0(ß.To see that the given map O is a diffeomorphism between neighbourhoods

of 0 and A just note that the inverse of T0O is given by the splitting

Lp(E, T*E <g> 0) = TA£ 8 *TA£

= dAW^{E,S) 0 imTo^e *dAW^(E,0) 0 *imT0</>

composed with the inverses of d^l^i.p^ -. and T0(ß.

Now observe that the choice of p > 2 for the Lagrangian submanifolds in

the above lemma is accidental. All connections A E £ are gauge equivalentto a smooth connection, and the L9-completion of £ n 4(E) is a Lagrangiansubmanifold in 40'9(E) for all q > 2. In fact, this simply is the restricted

(q > p) or completed (q < p) £]'9(E)-bundle over M.

44 CHAPTER 4. LAGRANGIANS

The main example

Suppose that G is connected and simply connected and that E = dY is the

boundary of a handlebody Y. (Again, the handlebody and thus its boundary

might consist of several connected components.) This together with the fact

that 7T2(G) = 0 for any Lie group G (see e.g. [B, Proposition 7.5]) ensures that

the gauge group Ç/lj3(E) is connected and that every gauge transformation

on E can be extended to Y.

Let p > 2 and let £Y be the Lp(E)-closure of the set of smooth flat

connections on E that can be extended to a flat connection on Y,

£Y := cl {A E 4flat(S) | 3Ä E 4flat(r) : I|s = A} C A°>P(Z).

This set has the following properties.

Lemma 4.3

(i) £Y = {u*(A\s) | A E Aut(Y),ue ^(E)}

(n) £Y C 4°'P(E) is a Lagrangian submanifold.

(in) £Y C 4flft(E) and £Y is invariant under the action of G1,p{^)-

(w) Fix any z E E. Then

£Y = {A E ^t(E) | pz(A) E Hom(7T1(F),G) C Hom(7r1(E), G)}.

So £Y obtains the structure of a Ç?]'P(E) -bundle over the g-fold productM = G x • • • x G ^ Hom(7T1(F), G),

£'P(E) ^ £Y ^ Hom(7T1(y), G).

Proof: Firstly, £Y C 4flft(E) follows from the fact that weak flatness is an

i7-closed condition for p > 2. The holonomy pz : 4^ft(E) —>• G x • • • x G

is continuous with respect to the Lp-topology. Thus for every A E £Y the

holonomy vanishes on those loops in E that are contractible in Y. On the

other hand, in view of theorem 3.1, every A E 4flft(E) whose holonomydescends to Hom(7Ti(F), G) can be written as A = v*A, where u E £]'P(E)and the holonomy of I e 4flat(S) also vanishes along the loops that are

contractible in Y. Thus A can be extended to a flat connection on Y and

45

smooth approximation of u proves that A E £Y. This proves the alternative

definitions of £Y in (iv) and (i). Then (iii) is a consequence of (i).To prove the second assertion in (iv) we explicitly construct local sections

of £Y. Let the loops ai,ßi,...,ag,ßg C E represent the standard generators

of 7Ti(E) such that oil,... ,ag generate 7Ti(F) and such that the only nonzero

intersections are at PI ßt.1 Now fix A E £Y. In order to change its holon¬

omy along at by some g EG close to 1, one gauge transforms A in a small

neighbourhood of ßt in E with a smooth gauge transformation that equals 11

and g respectively near the two boundary components of that ring about ßt.That way one obtains a smooth local section (ß : V —y £Y defined on a neigh¬bourhood V C 0s of 0, such that 0(0) = A and pz o <ß : V —y Hom(7Ti(F), G)is a bijection onto a neighbourhood of pz(A). This leads to a bundle chart

^ .

#*(E) x^ £Y

(u,v) i—y u*(ß(v).

Note that for smooth A E £Y fl «4(E) the local section 0 constructed above

in fact is a section in the smooth part £Y n 4(E) of the Lagrangian. Us¬

ing these bundle charts one also checks that £Y C 4°'P(E) is indeed a Ba¬

nach submanifold. A submanifold chart near *&(u,v) E 4°'P(E) is given by

(£, w) (->• vl>(exp(£)«, v + w) + *T(M;t,)\I/(£, w). As in lemma 4.1 one checks

that this is a local diffeomorphism.To verify the Lagrangian condition it suffices to consider uj on TAAQ,P(T)

for smooth A E £Y. This is because both u and £Y are invariant under

the gauge action. So pick some A E £Y fl 4(E) and find A E Aü&t{Y) such

that A = 4|s- Let a,ß E TA£Y, then by the characterization of £Y in (i)we find £, £ E W^façf) and paths Äa,Ä? : [-1,1] -+ 4flat(F) such that

i«(0) = 1^(0) = I and

« = dA£+£L0i*(S)|S, 0 = dX+£1^00 Is-

Now firstly Stokes' theorem on E with <9E = 0 proves

c^d^d^C) = lim f(dAeAdAC) = lim [ d{CAdAC) = 0.

Here we have used smooth VF^-approximations £v and C1" of £ and £ respec¬

tively.

1tti (S) is the quotient of the free group generated by elements ol\ , /?i,..., ag, ßg by the

relation aißia^ ß± .. .agßga~lß~l = 1, whereas tti(F) is the free group generated by

ai,...,ag.

46 CHAPTER 4. LAGRANGIANS

Similarly, one obtains a;(d/i£, ^A^|s) = 0 and u(-^Âa\-s,dA() = 0 since

dA(^ia|s) = £FAa\s = 0. Finally, Stokes' theorem with dY = E yieldsdue to FAa,s = 0 for all s

«>(<*> ß) = l(£Äa\^isÄßk) = fY^isÄa^isÄß)= fa £** ^îsA')-fY( îAa A £FÄ0 ) = 0.

This proves that co\tacy = 0 and recalling (4.2) one moreover sees that Ta£y

and *TA£Y are L2-orthogonal. In fact, we even have the topological decom¬

position LP(E, T*E ® 0) = TA£Y © *Ta£y, and this proves the Lagrangian

property of £Y. To see that this direct sum indeed exhausts the whole space

consider the Hodge type decomposition as in the proof of lemma 4.1,

i7(E, T*E ® 0) = dAW^(E, 0) 8 *dAT^1'p(E, 0) © hA.

Here we have dim fiA = 2g dimG, and we have already seen that £Y is

a C/]'p(E)-bundle over the (g dim G)-dimensional manifold Hom(7Ti(F), G).So dJ4T47z1'p(E,0) C Ta£y is the tangent space to the fibre through A, and

then for dimensional reasons Ta£y © *Ta£y also exhausts Ka and thus all

of Lp(E,T*E®0). D

Chapter 5

Cauchy-Riemann equations in

Banach spaces

In this chapter we give a general regularity result for Cauchy-Riemann equa¬

tions in complex Banach spaces with totally real boundary conditions. One

component of the boundary value problem (1.2) will take this form, and the

regularity result for this component will be the key point of the proof of

theorems A and B.

So consider a Banach space X with a complex structure J G EndX, i.e.

such that J2 = —11. Let £ C X be a totally real submanifold, i.e. a Banach

submanifold such that X = Tx£ © JTX£ for all x E £.

Example 5.1 In our application the Banach space will be 4°'P(E) with the

complex structure given by the Hodge operator J = * on 1-forms for any

metric on E. Then J is compatible with the symplectic structure u on

4°'P(E) given by (4.1), since u(-, J-) is an L2-inner product on the 1-forms.

Hence any Lagrangian submanifold £ C 4°'P(E) is totally real.

Recall that the gauge invariant Lagrangian submanifolds £ C 4flft(E)in the previous chapter were modelled on W^'p(E,0) © Rm. We will use a

similar assumption for the general totally real submanifolds £ C X.

Assumption: Throughout this chapter we suppose that the totally real

submanifold £ C X is - as a Banach manifold - modelled on a closed sub-

space Y C Z of an Lp-space Z = LP{M, Rm) for some p > 1, m E N, and a

closed manifold M.

47

48 CHAPTER 5. CAUCHY-RIEMANN EQUATIONS

Example 5.2

(i) Every finite dimensional space Rm is isometric to the subspace of con¬

stants in D>(M, Rm) for Vol M = 1.

(ii) The Sobolev space Wt,p(M) (and thus every closed subspace thereof)is bounded isomorphic to a closed subspace of LP(M, Rm).

To see this, choose vector fields X1;..., Xk E T(TM) that span T^M

for all x E M. Then the map u \-y (u,VXlu,... ,VX u) running

through all derivatives of u up to order I gives a bounded isomor¬

phism between Wl,p(M) and a closed subspace of LP(M, Rm) =: Z for

m = 1 + k + ... + ki. Estimates in Z then also yield IT^-estimates

since the norms are equivalent.

(iii) Finite products of closed subspaces in Lp(Mi,Rmi) are isometric to a

closed subspace of W{[\Mh R1^}).

Now let M be the half space

H:= {(s,t) ER2 J t > 0}.

Let Q C H be a compact 2-dimensional submanifold, i.e. Q has smooth

boundary that might intersect dM = {t = 0}. We consider a map u : Q —y X

that solves the following boundary value problem:

f dsu + Jsjdtu = G, , .

\ u{s,0) e£ V(s,0)EdQ.{ '

Here J : Q —y End X will be a suitably regular family of complex structures

on X. Now assume that u,G E Wk'q(Q, X) for some k E N and

„=1 p ;fc^2'q '

\ 2p ; k = 1.

In both cases the following theorem then gives II/fe+1'p-regularity for u. Here

and throughout the interior of Vt is defined with respect to the topology of

H, so int Q still contains dQ fl dM.

49

Theorem 5.3 Fix 1 < p < oo and a compact subset K C mtVt. Let

Uq E C°°(fl,X) be such that u0(s,0) E £ for all (s, 0) G dQ, and let J0 £

C°°(f2,EndA) 6e a smooth family of complex structures on X. Then there

exists a constant Si > 0 and for all k E N t/jere ex«s£ constants S2 > 0 and C

SMc/i t/iat £/ie following holds:

Let J E Ck+1(Q, End X) 6e a family of complex structures on X. Supposethat u,G E Wk,q(Q,X) (with q as above) solve (5.1) and that

\\u — Mo||L°°(n,X) < ^i, || J — Jo\\ck+1(n,EndX) < ^2-

Then u E Wk+l*{K,X) and

\\u — uQi\\wk+lp{K,x) < C[l + \\G\\Wk,q(ci!X) + ||« — uo\\Wk,q(ntXA.

Note that the Uq in this theorem is not a solution of the equation. It

only satisfies the boundary condition and will be required for the choice of

coordinates near £ in the proof.Theorem 5.3 will be essential for the nonlinear regularity and compact¬

ness results in chapter 6. For the Fredholm theory in chapter 7 we will

moreover need the following regularity and estimate for the linearization of

the boundary value problem (5.1). In order to state the corresponding weak

equation we introduce the dual operator J* E End X* of the complex struc¬

ture J E EndX, where X* denotes the dual space of X.

Theorem 5.4 Fix 1 < p < oo and a compact subset K C intfi. Fix a

path x E C°°(R, £) in £ and let J E C°°(f2,EndX) be a family of complexstructures on X. Then there is a constant C such that the following holds:

Suppose that u E IPi^l^X) and that there exists a constant cu such that

for all-iß EC°°(n,X*) with supp iß C intfi andiß(s70) E (J(s, 0)Tx(a)Z;)± forall (s, 0) G dQ

(u7dsiß + dt(J*iß)) < cu\yv\\LP*(n,x*'n

ThenuEW^p{K,X) and

IM|iyi,P(^,X) < C(cu + ||-u||LP(n,X))-

Corollary 5.5 Under the assumptions of theorem 5.4 there exists a con¬

stant C such that the following holds: Suppose that u E W1,p{Vt,X) satisfies

u(s, 0) G Tx(s)£ for all (s, 0) G dVL, then

\\u\\w^ï>{k,x) < C(\\dsu + Jdtu\\Lv(ÇL,X) + |M|LP(n,x))-

50 CHAPTER 5. CAUCHY-RIEMANN EQUATIONS

Proof of corollary 5.5:

Let u E W^p(fl,X) and iß E C°°(Q,X*) such that supp^ C int^ and with

the boundary conditions u(s, 0) G Tx^£ and iß(s,Ö) E (J(s, O^^C)1- for

all (s, 0) G dCl. Then one obtains the weak estimate, where the boundaryterm vanishes,

{u,dsiß + dt(J*iß))n

/ (dsu + Jdtu, iß) - l (Ju,iß)Jn Jdnndu

< \\dsu + Jdtu\\LP(a,x)\\^ß\\Lv*(n,x*)-

This holds for all iß as above, so the estimate follows from theorem 5.4. D

Remark 5.6 Theorem 5.4 and corollary 5.5 also hold when one considers

compact domains K C int Vt C S"1 x [0, oo) in the half cylinder and a loop

xEC°°(S\£).To see this, identify S1 = M/Z, identify K with a compact subset if'cH

in [0,1] x [0, oo), and periodically extend x and m for s G [—1, 2]. Then u is

defined and satisfies the weak equation on some open domain il! C H such

that K' C intfy, so the results for HI apply.

Recall that the totally real submanifold £ is modelled on the Banach

space Y. The idea for the proof of theorem 5.3 is to straighten out the

boundary condition by going to local coordinates in Y x Y near u0 E X such

that Y x {0} corresponds to the submanifold £ and the complex structure

becomes standard along Y x {0}. For theorem 5.4 one chooses M-dependentcoordinates for X that identify Y x {0} with Tx(s)£ along the path x : R —y £.

Then the boundary value problem (5.1) yields Dirichlet and Neumann

boundary conditions for the two components and one can use regularity re¬

sults for the Laplace equation with such boundary conditions. However, there

are two difficulties. Firstly, by straightening out the totally real submanifold,the complex structure J becomes explicitly dependent on u, so one has to

deal carfully with nonlinearities in the equation. Secondly, this approach re¬

quires a Caldéron-Zygmund inequality for functions with values in a Banach

space. In general, the Caldéron-Zygmund inequality is only true for values in

Hilbert spaces. In our case we only need the i7-inequality for functions with

values in Lp-Sobolev spaces. In that case, the Caldéron-Zygmund inequality

holds, as can be seen by integrating over the real valued inequality. This will

be made precise in the following lemma. We abbreviate A := d*d, denote by

51

Z* the dual space of any Banach space Z and by ( -,• ) the pairing of Z and

Z*. The Sobolev spaces of Banach space valued functions considered below

are all defined as completions of the smooth functions with respect to the

respective Sobolev norm. Moreover, we use the notation

Cf{i\ Z*) := {iß E C°°(Q, Z*) I iß\dn = 0},

C?(tl,Z*) := {iß E c°°(n,z*) I g\m = 0}.

Lemma 5.7 Let fl be a compact Riemannian manifold with boundary, let

1 < p < oo and k E N. Let Z = LP(M) for some closed manifold M. Then

there exists a constant C such that the following holds.

(i) Let f E Wk~1'p(fl, Z) and suppose that u E Wk>p(Q, Z) solves

f{u,Aiß) = f(f,iß) VißECr(Vl,Z*).Jvl Ja

Then u E Wk+lj,(ü, Z) and \\u\\Wk+i,P < C\\f\\Wk-i,P.

(ii) Let f E Wk-^p{i\ Z), g E Wk>p(Q, Z), and suppose that u E Wk'p(Q: Z)solves

f(u,Aiß) = f{f,iß)+ f {g,iß) \/ißEC(Q,Z*).Jn Ja. Jan

ThenuEWk+1'p(Q,Z) and

\\u\\Wk+i,P < c(11/11 w*-1^ + IMIwfc'P + IMU")-

(iii) Suppose that u G i7(Q, Z) and there exists a constant cu such that

u- AnißQxM

< C„ i^(q,z*) V^eCf(fixM).

Then u G W1,p(fl, Z) and IMIiyi^^z) < Ccu.

(iv) Suppose that u G i7(Q, Z) and there exists a constant cu such that

u- Aniß < c„ w^v*{n,z*) Vip G C?(Q x M).'QxM

ThenuE W^P(Q,Z) and

||w||wi,P(n,z) < C[cu + ||m||lp(îî,z))-

// moreover fn u = 0 then in fact |M|wi,p(n,z) — Ccu

52 CHAPTER 5. CAUCHY-RIEMANN EQUATIONS

The key to the proof of (i) and (ii) is the fact that the functions / and g

can be approximated not only by smooth functions with values in the Banach

space LP(M), but by smooth functions on fl x M.

Lemma 5.8 Let fl be a compact manifold (possibly with boundary), let M

be a closed manifold, let 1 < p,q < oo7 and k,£ E N0. Then C°°(fl x M) is

dense in Wk'q(fl,We'p(M)).

Remark 5.9

(i) A function u E Wk'q(fl, Wl'p(Mj) with zero boundary values u\ga = 0

can be approximated by uv E C°°(fl x M) with uv\qçixm = 0.

This can be seen by first approximating in C°°(fl, Wi,p(M)) with zero

boundary values and then mollifying on M as in lemma 5.8. In case

k = 0 the boundary condition is meaningless, but the approximationwith zero boundary values can be done elementary by cutting off in

small neighbourhoods of the boundary. For k > 1 consider a local chart

of fl in [0,1] xRn such that {t = 0} corresponds to the boundary, where

t denotes the [0, Incoordinate. Let / G Wk'q([0,1] x W1, Z) for any vec¬

tor space Z with /|t=o = 0 and compact support. Let a£ be mollifiers

on Rn as in the proof of lemma 5.8 below, then f£(t, •) := a£ * f(t, •)defines fe E C°°(Rn,Wk>q([0,l],Z)) for all e > 0. One checks that

ll/e ~~ /||vrfe.8([o,i]xK'1) ""* 0 as e —y 0. We choose the a£ with compact

support, then the f£ are also compactly supported and hence have fi¬

nite Wk>q([0,1], W^q(Rn))-noTm for any £ E N. Moreover, note that still

/e|t=o = 0. In order to approximate f£ with zero boundary values one

chooses t = k, then lemma 5.8 gives a smooth approximation gv —y dtf£in the W^-^QO, 1], Wk'q{Rn))-no?m. Now f?(t,x) := f*g"{r,x)dT de¬

fines functions in C°°([0,1] x W1, Z) that vanish at t = 0 and approxi¬mate f£ in the jy^QO, 1], I47/:''7(Mn))-norm, which is even stronger than

the IT^-norm on [0,1] x Rn.

(ii) If £p > dim M and z E M, then a function u E Wk>q(fl, W^P{M)) with

u(-, z) = 0 G Wk'q{fl) can be approximated by uv E C°°(fl x M) with

u"(;z) = 0.

Indeed choose an approximation by vv E C°°(r2 x M), then uu(-,z) —y 0

in Wk,q(fl) since the evaluation at z is a continuous map Wi,p(M) —y R.

Now uv-uu(;z)eC00(nxM) still converges to u in Wk'q(fl, Wl*{M))but it vanishes at z.

53

Proof of lemma 5.8:

By definition C°°(fl, Wl>p(M)) is dense in Wk>q(fl, We'p(M)): so it suffices to

fix g E C°°(^, Wl*{M)) and show that in every Wk>q(Q, TT^'p(M))-neighbour-hood of g one can find a g E C°°(fl x M). Firstly, we prove this in the case

k = 0 for closed manifolds M as well as in the following case (that will be

needed for the proof in the case k > 1): M = Rn, g is supported in Q x V

and g is required to have support in fl x U for some open bounded domains

Pel" such that V C U.

Fix S > 0. Since Q is compact one finds a finite covering Q = |JÎ=1 U% by

neighbourhoods Ut of x% E fl such that

\\g(x) - g(xt)\\wi,P{M) < | Vie Ut.

Next, choose g% E C°°(M) such that ||g, — g^) || w^,p(m) < §• In the case

M = Rn one has suppg(xj) C V and hence can choose gt such that it is

supported in U (e.g. using mollifiers with compact support). Then choose a

partition of unity X^=i 4 = 1 by ^ £ C°°(n, [0,1]) with suppçi», C £/,. Now

one can define g G C°°(Q x M) by

JV

g(x, z) := ^2 Mx)9i(z) Mx Efl,z G M.

î=i

In the case M = Rn this satisfies supp g C fl x U as required. Moreover,

ll~ — \\q—I IIV^ tk ( —

\\\q»9 9\\Lq^we,p^M))

— / ||Z^î=i <Pi\9t 9)\\wt,p(M-)

< / G,=i 0« •

suP*e^ lift - tfOOIIw^M))97n

< Sq = 5q\olfl.Jn

Thus we have proven the lemma in the case k = 0. For k > 1 this method

does not work since one picks up derivatives of the cutoff functions <f>%. In¬

stead, one has to use mollifiers and the result for k = 0 on M = Rn.

So we assume k > 1, fix g G C°°(ft, W^'P(M)) and pick some S > 0. Let

M = Uî=1 <l\(£/i) be an atlas with bounded open domains Ut C Rn and charts

$,:[/, ->• M. Let K C F, C t/, be open sets such that still M = J^1 ^l{Vl).Then there exists a partition of unity ^=1 ißto§;1 = lbyißtE C°°{Rn, [0,1])such that supp-^j C Vl. Now g = ^î=15,î°(idn x Q'1) with

9t(x,y) = ifrt{y) g{x,®%(y)) VxeQ,ye U%.

54 CHAPTER 5. CAUCHY-RIEMANN EQUATIONS

Here gt G C°°(fl, We>p(Rn)) is extended by 0 outside of supp& C Q x Vt,and it suffices to prove that each of these functions can be approximated in

Wk'q{fl, Wl'p(Rn)) by g% G C°°(£l x Rn) with supp& C fl x Ut. So drop the

subscript i and consider g G C°°(fl, Wl'p(Rn)) that is supported in fl x V,where V7U CRn are open bounded domains such that V C U.

Let a£(y) = e~na(y/e) be a family of compactly supported mollifiers for

e > 0, i.e. a G C°°(Mn, [0,oo)) such that suppa C #i(0) and Ja = 1. Then

for all s > 0 define & G C°°(f2 x Rn) by

&(z, y) := [a£ * <?(x, •)](?/) Vx G fl, y G Mn.

Firstly, supp<7£ C Be(0), so for sufficiently small e > 0 the support of g£ lies

within fl x U. Secondly, we abbreviate for j < k, m < £

fj,m := yJnV^g G C°°(n,LP(Rn)),

which are supported in fl x V. Then

\\9e — 9\\wl',q(çi,We'P(Rn)) = 2_^ / H^nl^e * 9{x: ') ~ 9{x, ')} \\w1>p(9P)]<k

Jn

<{e+1)vJ2z2 W^* hm{x, ) ~ fj,ru(Xr)\\qLP{Rny3<k m<lJçi

Now use the result for k = 0 on M = Rn (with values in a vector bundle) to

find fj,m EC°°(fl x Mn) supported in fl x U such that

\\fj,m — fj,m\\L<i(n,LP(M.n)) < &

Then for all x G fl and sufficiently small s > 0 the functions cr£ * fj,m{xi ') are

supported in some fixed bounded domain U' C Rn containing U. Moreover,the f],m are Lipschitz continuous, hence one finds a constant C (dependingon the fhmi i-e- on 9 and 5) such that for all x G fl

\<Je * Jj,m\X7 ') Jj,m\xi \LP(Rn)

'U-

<

ànyOe{y' - y){fj,m(x,y') - f3,m{x,y))àny'

I (/ cr£(y'-y) sup \f3,m{x, y') - f3,m{x, y)\ dny'\ dnyJu' yJM.n \y-y'\<e

<Vo\U'{Ce)p.

55

Now use the fact that the convolution with a£ is continuous with respect to

the i7-norm, ||a£ * f\\p < \\f\\p (see e.g. [Ad, Lemma 2.18]) to estimate

/ \\a£ * J3,m\x: ') ~ f3,m\x, ')\\]j>(^n^J il

— / I Ir6 * {jJ,m\xj ') ~ J3,m{xi ')) ||ip(K7i) ' \\j3,m\x-> ') ~ J3,m\xi 'J ||ip(Rn)

II~

"II \q+ ||°V * J3,m\xi ')

~

J3,m\xi ')\\LP(K.n))

<2-3i/Jim-/J>m||^(niLP(RB))+3«Voin(VolE0*(C7^ < 3-3W

Here we have chosen 0 < e < C~l(Vol fl)~« (Volt/)-p S. Thus we obtain

ll^e - 9\\wi°«(a,wt*(R»)) < S(£+l)p(3(k + l)(£ + l))^6.

This proves the lemma.

In the case q = p this lemma provides the continuous inclusion

Wk'p(fl,W£'p{M)) C Wl'p(M,Wk'p(fl))

since the norms on these spaces are identical. 1Moreover, for p = q and

k = £ = 0 the lemma identifies U>(fl, U'(M)) = U'(fl x E) as the closure of

C°°(fl x M) under the I7-norm.

Proof of lemma 5.7 (i) and (ii) :

We first give the proof of the regularity for the inhomogenous Neumann

problem (ii) in full detail; (i) is proven in complete analogy - using the reg¬

ularity theory for the Laplace equation on R-valued functions with Dirichlet

boundary condition instead of the Neumann condition.

So let / G Wk-^p(fl,Z): g E Wk'p(fl, Z), and choose approximating

sequences f\g% E C°°(fl x M) given by lemma 5.8. Note that testing the

weak equation with iß = a for all a E Z* yields the identity /n / + fdn g = 0.

Thus h% := jn f% + Jdn g% —>• 0 in Z as i —y oo, so one can replace the /* by

/* - hl/Xo\fl E C°°(M, Z) to achieve

f n,y)+ [ 9t(;y) = 0 VyEM,iEN.Jn Jan

1The spaces are actually equal. The proof requires an extension of the approximation

argument to manifolds with boundary. We do not carry this out here because we will onlyneed this one inclusion.

56 CHAPTER 5. CAUCHY-RIEMANN EQUATIONS

Now for each y E M there exist unique solutions ul(-,y) E C°°(fi) of

A««(-,y) = /'(, a),

/n«f(-.y) = o.

For each of these Laplace equations with Neumann boundary conditions one

obtains an //-estimate for the solution, see proposition 3.5 and e.g. [We,Theorem 3.1] for the existence. The constant can be chosen independentlyof y E M since it varies continuously with y and M is compact. Then

integration of those estimates yields (with different constants C)

n 1 u» / ii 1 iip

II" llTv*+i.p(n,z) / \\a \\wk+1'P(n)

< / C(l|.T||iyfc-1^(n) + H^llw^n))JM

< C(l|.T||wfc-1'î>(fi,z) + ||â''||w*>p(n,z)) •

Here one uses the crucial fact that W(fl, U>(M)) C LP(M, LP(fl)) with iden¬

tical norms. (Note that this is not the case if the integrability indices over fl

and M are different.) Similarly, one obtains for all i,jEN

\\u% - u3\\Wk+i,P^Z) < C(\\P - /J||i4/fc-i,P(n,z) + \\9l - 9J\\w^p(n,z))-

So u% is a Cauchy sequence and hence converges to some ü E Wk+1,p(fl, Z).Now suppose that u E Wk,p(fl, Z) solves the weak Neumann equation for /and g, then we claim that in fact u = ü + c E Wk+1,p(fl, Z), where c G Z is

given by

c(y) :=

vô~m / M''y) ~ û(''y^ Vy E M-

In order to see that indeed c G LP (M) = Z and that for some constant C

one has ||c||lp(m) < C(IMUp(n,z) + II^Ulp^z)) n°fe that lemma 5.8 yields the

continuous inclusion Wk'p(fl,'Lp(M)) C 'lp(M, Wk'p(fl)) C D>(M, L1 (fl)).To establish the identity u = ü + c, we first note that for all (ß E C°°(M) C Z*

(ü + c — u, (ß) = / (ß (c— (ü — u)j= 0.

Jn Jm ^ Jn'

Next, for any <ß E C°°(fl x M) let

57

Then one finds iß E C^(fl x M) such that <f> = A^iß + cßc,. (There exist uniquesolutions iß(-, y) of the Neumann problem for (ß(-, y) — (ßo(y), and these depend

smoothly on y G M.) So we find that for all <f> E C°°{fl x M), abbreviating

An = A

(u-ü- c, cß) = (u, Aiß ) - {ü + c, Aiß )+ (u — ü — c, (ß0)Jn Jn Jn Jn

f-Au\lß) + [ (g-^,iß)) = 0.Jas

lim

'an

This proves u = ü + c E Wk+1'p(fl, Z) and the estimate for ul yields in the

limit

||w||wr*+1J>(n,z) < IIwII^^+lp^.z) + (Vo1J1)p||c||lp(m)< Cdl/llw^-Lp^.z) + IMIw^n.z) + IMU"(n,z))-

This finishes the proof of (ii), and analogously of (i).

Proof of lemma 5.7 (iii) and (iv) :

Let u E W(fl, Z) be as supposed in (iii) or (iv), where Z = W(M) and thus

Z* = Lp*(M). Then we have u E LP(fl x M) and the task is to prove that

dftu also is of class Confix M. So we have to consider fnxMu d^r for

t EC$°(flxM, T*fl) (which are dense in Lp* (fl x M, T*Q)). In the case (iii)one finds for any such smooth family r of 1-forms on fl a smooth function

iß E Cf(fl x M) such that d^r = A^iß. Then there is a constant C such

that for all y E M (see proposition 3.5)

'(,y)ll w^p* < C\\Aniß(;y)\\{wi,Pr < C\\r(;y)\\f

In the case (iv) one similarly finds iß E C^(fl x M) such that d^r = A^iß and

IW'^llw1^*—

C llr(')?/)llp* f°r all y G M and some constant C. (Note that

In ^nr — 0 since r vanishes on dfl x M and we have used e.g. [We, Theorems

2.2,2.3'].) In both cases we can thus estimate for all r G C%°(fl x M,T*fl)using the assumption

/ «

JnxMd^r /

JnxM

u- Aniß < c„

M

P

w^-'P'in)

< Cc, (/\JM

\hp*{n)

p*

< Ccu\\tu\\l \\Lp (nxM)-

58 CHAPTER 5. CAUCHY-RIEMANN EQUATIONS

Now in both cases the Riesz representation theorem (e.g. [Ad, Theorem 2.33])asserts that JnxMu d^r = JnxM / r for all r with some / G W(fl x M).This proves the /^-regularity of d^u and yields the estimate

||dn«||LJ>(nxM) < Ccu.

In the case (iii), one can moreover deduce u\qq = 0. Indeed, partial integra¬tion in the weak equation gives for all iß E Cg°(fl x M)

u

anxM

a±dp

u Atfß - / ( dnu , dn-0 )M JnxM

< (cu + ||dnM|Up(nxM))||^IU1,P*(nxM)-

For any given g E C°°(dfl x M) one now finds iß E C°°(fl x M) with iß\9n = 0

and |jj = g, and these can be chosen such that H^Hiy1^* becomes arbitrarilysmall. Then one obtains fdnxMug = 0 and thus u\on = 0. Thus in the

case (iii) one finds a constant C such that

\u\wl<p{n,z)

\u\M

w^p(n)<C f \\dnu\\pLP{n) = C'\\dnu\\

JMLP{nxM)i

which finishes the proof of (iii).In the case (iv) with JQ u = 0 one also has a constant C such that

||«(-,2/)||iy1.p(f2) < C'\\dnu(-,y)\\LP{n) for all y E M and thus

IMIw^n.z) < C||dn?x||LP(nXM) < C'Ccu.

In the general case (iv) one similarly has

\u\wl'P{n,z) l|dnwlliP(nxM) \u\

LP(nxM)< {Ccu + ||«|Up(n,z))P-

D

The proof of theorem 5.3 will moreover use the following quantitative

version of the implicit function theorem. This is proven e.g. in [MS2, Propo¬sition A.3.4] by a Newton-Picard method. (Here we only need the special

case Xq = x\ = 0.)

59

Proposition 5.10 Let X and Y be Banach spaces and letU C Y be a neigh¬bourhood of 0. Suppose that f : U —> X is continuously differentiable map

such that d0/ : Y —y X is bijectwe. Then choose constants c > ||(do/)_1||and 5 > 0 such that B$(0) C U and

K/-do/||<£ VyEB0(0).

Now if ||/(0)|| < £; then there exists a unique solution y E B$(0) of f(y) = 0.

Moreover, this solution satisfies

ll!/ll<2c||/(0)||.

Proof of theorem 5.3 :

For every (s0, 0) G fl one finds a Banach manifold chart (j) : V —y £ from

a neighbourhood V C Y of 0 to a neighbourhood of ^(0) = Uq(s0,0) =: x0.

Choose a complex structure J G EndX, then one obtains a Banach subman¬

ifold chart of £ C X from a neighbourhood W C Y x Y of zero to a ball

B£(xq) C X around x0,

q.W ^ B£(x0)

(vuv2) i—y 4>{vi) + JdVl(ß(v2).

To see that this is indeed a diffeomorphism for sufficiently small W and

e > 0 note that D := d(o,o)@ = d0^ © Jd0(ß. Here d0çi> : Y —>• TXo£ is an

isomorphism and so is the map TXo£ x T^/Z —>• X given by the splittingX = ±x0£ © J lXq£.

Now let the complex structure J vary in a sufficiently small neighbour¬hood of Jo(so,0) G End A" such that ||ö_1|| < c with a uniform (i.e. J-

independent) constant c. Since (ß is smooth one then also finds a uniform

constant 5 > 0 such that ||d„0 — D|| < ^ for all v = (vi,v2) G i^(0) and

-B<5 (0) C W. So one can apply the quantitative implicit function proposi¬tion 5.10 to the maps / = O — x for varying J and x E X if \\x — x0\\ = ||/(0)||is sufficiently small. This yields uniform constants e, ô > 0 and C such that

O-1 : B£(xq) —y Bs(0, 0) C Y x Y is uniquely defined for all complex struc¬

tures J in the neighbourhood of Jo(sq, 0), and moreover

IIO-^Hzxz < C\\x - xQ\\x Mx E B£(x0). (5.2)

60 CHAPTER 5. CAUCHY-RIEMANN EQUATIONS

(Recall that F is a closed subspace of the Banach space Z, so the norm on

Y is induced by the norm on Z.) In particular, the uniform estimate (5.2)holds for all J = JStt if (s, t) E U for a neighbourhood U C fl of (s0, 0) and

provided that J E Ck+1(fl, EndX) satisfies the assumption \\J — Jo||c*+1 < ^2

for sufficiently small S2 > 0. Thus one obtains a Cfc+1-family of chart maps

for (s, t) E Z7,

Qs,t : Y x Y D Ws,t ^ B£(x0).

Next, choose U even smaller such that Uq(s, t) E Bl (x0) for all (s, t) E U and

let 5i = |. Then every u E Wk'q(fl,X) that satisfies \\u — Uo\\L°°(n,x) < ^1

can be expressed in local coordinates,

u{s,t) = Q8tt{v{s,t)) V(s,t)EU,

where v E Wk,q(U, Z x Z). This follows from the fact that the composition

of the Cfc+1-map O-1 with a Wk'q-msip u is again VTfc'9-regular if kq > 2 (seee.g. [We, Lemma B.8]). Moreover, v actually takes values in W C Y x Y.

Integration of (5.2) together with the fact that all derivatives of O-1 up to

order k are bounded yields the estimate

IMIwM(c/,zxz) < C\\u - u0\\wk,i(u,x) < C81. (5.3)

Here and in the following C denotes any constant that is independent of the

specific choices of J and u in the fixed neighbourhoods of J0 and u0.

In these coordinates, the boundary value problem (5.1) now becomes

/ dsv + Idtv = /, , ,

\ v2(s,0) = 0 VsGM.K '

with v = (vi,v2) and

f = (dv&)-\G - dsQ(v) - JdtQ(v)) E Wk>q(U,YxY),I = (dyQ)-1 Jdv@ E Wk'q(U, End(Y x Y)).

Note the following difficulty: The complex structure / now explicitly depends

on the solution v of the equation (5.4) and thus is only IT^-regular. This

cannot be avoided when straightening out the Lagrangian boundary condi¬

tion. However, one obtains one more simplification of the boundary value

61

problem: O was constructed such that one obtains the standard complexstructure along £. Indeed, for all (s, 0) G U using that J2 = — 1

l(s,0) = (d(„1;0)0)"1Jd(l,l!o)0 = (dvi(ß® Jdvicß)~ J{dvi(ß © Jdvi(ß)

= (!"!)-*Moreover, one has the following estimates on U:

II T\\ •e"' cW1 ||w*>« — t-y)

ll/II^M < C(I|C||wm + II« - WoIIvtm)-

So for every boundary point (s0, 0) G f2n<9H we have rewritten the boundaryvalue problem (5.1) over some neighbourhood U C fl. Now for the compact

set K C fl one finds a covering K C V U Uî=i ^ by finitely many such

neighbourhoods Ut at the boundary and a compact domain VcO\dfl away

from the boundary. Note that the Ut can be replaced by interior domains

U% (that intersect dUl only on dM) that together with V still cover K. We

will establish the regularity and estimate for u on all domains Ul near the

boundary and on the remaining domain V separately. So firstly consider

a domain U% near the boundary and drop the subscript i. After possibly

replacing U by a slightly smaller domain one can assume that U is a manifold

with smooth boundary and still U fl dU C dM. The task is now to prove the

regularity and estimate for u = Q o v on U from (5.4).Since 0S)i : Y x Y —y X are smooth maps depending smoothly on

(s,t) E U, it suffices to prove that v E Wk+1'p(U, Z x Z) with the accordingestimate. (One already knows that v takes values - almost everywhere - in

Y x F, so one automatically also obtains v E Wk+1,p(U, Y x Y).) For that

purpose fix a cutoff function h E C°°(H, [0,1]) with h = 1 on U and h = 0

on H\ U. Moreover, this function can be chosen such that dth\t=o = 0. Note

that h = 0 on dU \ dM, so hv2 satisfies the Dirichlet boundary condition

on dU. Indeed, we will see that hv2 E Wk'p(U, Z) solves a weak Dirichlet

problem.

In the following Y* denotes the dual space of Y and ( -,• ) denotes the

pairing between Y and Y* as well as the pairing between Y x Y and Y* x Y*.

Let I* E Wk'q(fl,End(Y* x Y*)) be the pointwise dual operator of/. Then

62 CHAPTER 5. CAUCHY-RIEMANN EQUATIONS

one has A = -(ds + dtL*)(ds - I*dt) + (dtI*)ds - (dsI*)dt and thus for all

4>EC°°(fl,Y* x Y*)

hA<ß = -(ds + dtI*)(ds - Pdt)(hcß) - (Ah)cß + 2(dsh)ds<ß + 2(dth)dtcß

+ (dtI*)ds(hcß) - (dsL*)dt(hcß).

Hence partial integration (using smooth approximations of v, /, and /) yieldsdue to (5.4)

f(hv,A<j>)Ju

= [(dsv + Idtv,(ds-Fdt)(h(ß))Ju

- f {(Ah)v + 2(dsh)dsv + 2(dth)dtv + h(dtI)dsv - h(dsL)dtv , <f> )Ju

+ / (Lv,(ds-Pdt)(h(ß)) + (h(dsL)v-2(dth)v,(ß)Jdunaw

= f ( h(-dsf + Idtf + (dtI)f - (dtI)dsv + (dj)dtv)Ju

- (Ah)v - 2(dsh)dsv - 2(dth)dtv , <j> )

+ / (h-If ,</>)+ [ (v,dt(h(ß)) + (Lv,ds(h(ß))JdUndW JdUndEL

= f(F,<ß)+ [ (H,<f>)+ [ (vudtihfà + dsihfc)). (5.5)Ju JdU JdUndW

Here we used the notation (ß = (0i, (ß2), the boundary condition v2\t=o = 0,and the fact that I\t=0 = I0. One reads off F = (Fi, F2) G Wk-lj>(U, Y x Y),H = (Hu H2) G Wk'p(U, Y xY), and that for some finite constants C

\\F\\]yk-i,P + ||Z/||yyfc,p < C (||/ ||]yk,q + ||/||vk*'<! 11/||tV*'« "I" H-» llTy^^lpllv^*'« j< CdlGU^/fe,«, + \\u — U0\\wk,q).

We point out that the crucial terms here are (dsI)dtv and (dtI)dsv. In the

case k > 3 the estimate holds with q = p due to the Sobolev embeddingWk-i,P . Wk-i,P ^ Wk-i,P_ In the case fc = i one only has L2p L2p ^ D>

and hence one needs q = 2p in the above estimate. In the case k = 2 the

Sobolev embedding W1,p W1,p ^->- W1,p requires p > 2. Let us make this

assumption for the moment and finish the proof of this theorem under the

63

additional assumption p > 2 in case k = 2. Then after that we will show

how an iteration of the k = 1-case of the theorem will give the required

IT^-regularity of (dsI)dtv and (dtL)dsv also in case 1 < p < 2. (Note that

this would follow from W2,p -regularity of u for any p' > 2.)Now in order to obtain a weak Laplace equation for v2 we test the weak

equation (5.5) with (ß = ((ßx,^) = (0,7r o iß) for iß G Cg°(U,Z*) and where

7T : Z* —y Y* is the canonical embedding. In that case, both boundary terms

vanish and one obtains for all iß E Cf(U, Z*)

f(hv2,Aiß) = f(F2,iß).Ju Ju

By lemma 5.7 (i) this weak equation for hv2 E Wk,p(U, Z) now implies that

hv2 E Wk+1>p(U,Z) and thus v2 E Wk+1>p(U,Z). Moreover, one obtains the

estimate

II^HjyH-i.p^z) < II^Hw^+lp^z) < C||-^2||wfc-i.P([/,Z)< C'(||G'||W/fc,<!(n,x) + \\u — uo\\wk^(n,x))-

To obtain a weak Laplace equation for v\ we have to test the weak equa¬

tion (5.5) with (ß = ((ßt,(ß2) = (7T o iß,0), where iß G C°°(U,Z*) such that

dttß\t=o = 0 in order to make the second boundary term vanish. One obtains

for all iß EC(U,Z*)

f(hVl,Aiß)= f(F1,iß)+ f (H,,iß).Jn Jn Jan

So we have established a weak Laplace equation with Neumann boundarycondition for hv\. Now lemma 5.7 (ii) implies that hv\ E Wk+1,p(U, Z), hence

vi E Wk+1,p(U,Z). Moreover, one obtains the estimate

|pi|lwAfc+1-p(ßr,z) ^ II^IItv*+1.p(;7,z)< Cdl-FLllw^-LP^Z) + ||i?l||wrfc,P(i7,z) + ll^lllw^'P^Z))< C'dlGHtf/M^x) + ||« — uo\\wk-i(n,x))-

This now provides the regularity and the estimate for u = Q o v on U as

follows. We have established that v : U ^ Z x Z is a, Wk+1,p-map that takes

values in W C Y x Y. All derivatives of O : fl x W —y X are uniformly

64 CHAPTER 5. CAUCHY-RIEMANN EQUATIONS

bounded on fl. Hence u E Wk+1<p(U,X) and

\\u — uo\\wk+1'P(Ü,X) ^ C(l + ||^||i4/fc+i,p(i/,x))< C(l + llGlliy*.«^,^) + llM _ uo\\wk>i(n,x))-

For the regularity of u on the domain Vcü\ dfl away from the boundary

one does not need any special coordinates. As for U, one replaces fl by a

possibly smaller domain with smooth boundary. Moreover, one chooses a

cutoff function h E C°°(M, [0,1]) such that h\v = 1 and that vanishes outside

of fl C H and in a neighbourhood of dfl. Then in the same way as for (5.5)one obtains a weak Dirichlet equation. For all <ß> E Cf{fl,X*)

f(hu,A(/>)= [(h(-dsG + JdtG + (dtJ)G - (dtJ)dsu + (dsJ)dtu)Jn Jn

- (Ah)u - 2(dsh)dsu - 2(dth)dtu, </>).

Note that X = YxYcZxZ also is bounded isomorphic to a closed

subspace of an Lp-space. So by lemma 5.7 this weak equation implies that

hu E Wk+1'p(fl,X), and thus u E Wk+1>p(V,X) with the estimate

||w||vpH-i,p(y,jr) < CdlGH^/M^x) + \\u — uo\\wk'0(n,x))-

Thus we have proven the regularity and estimates of u on all parts of the

finite covering K C VU|Jz=1 U%. This finishes the proof of the theorem under

the additional assumption p > 2 in case k = 2.

Finally, let u E W2'p(fl,X) and G E W2'p(fl,X) be as supposed for

1 < p < 2. Then the task is to establish W2,p -regularity and -estimates for

some p' > 2. This follows from the following iteration.

One starts with I/F2'Po(il0)-regularity and -estimates for pQ = p E (1,2)on fl0 = fl. (In case p = 2 one chooses a smaller value for p.) Now as longas Pi <

2X" < 2 and Ü, C H is compact one has the Sobolev embeddings1 Pi 2p,

W2'p(fl) m- W ,2-^(flz) and W2^(flt) m- L2-"' (flA. So choose a compact

submanifold fll+i C HI such that K C intfll+i and fll+\ C intslj. Then the

theorem in case k = 1 with p replaced by ^— gives regularity and estimates

in IT^'^+^f^+i) for pl+i = ^7—. One sees that the sequence (pz) grows at a

rate of at least 7^— > 1 until it reaches pN > -0- after finitely many steps.

A further step of the iteration with pN = -^- then gives W2,PN+1 -regularity

and -estimates with p^+i = -^r > 2 on fl^+i '= K. This is exactly the

regularity for u that still was to be established.

65

Proof of theorem 5.4 :

The Banach manifold charts near the path x : 1

path of isomorphisms (ßs : Y ^y Tx(s)£ f°r all s E

family of complex structures J E C°°(fl, EndX) these give rise to a smooth

family O G C°°(Q,Hom(F x Y, Xj) of bounded isomorphisms

£ give rise to a smooth

Together with the

9s.t

Y xY

(zi,z2) (ßs(zi)

X

- JSyt<ßs(z2).

The inverses of the dual operators of 0Sji give a smooth family of bounded

isomorphisms 9' G C°°(fl,}iom(Y* x Y*',X*)) ,

©'..t == (ej,*)"1 :TxfA X*.

One checks that for all (s, t) E fl

e;},Js,tQs,t = (Î"J) =: h End(rxF).

Next, after possibly replacing fl by a slightly smaller domain that still con¬

tains K in its interior, one can assume that fl is a manifold with smooth

boundary. Then fix a cutoff function h E C°°(H, [0,1]) such that h\x = 1

and supp h C fl, i.e. h = 0 near dfl \ dM. Now let u E Lp(fl, X) be given as

in the theorem and express it in the above coordinates as u = Q o v, where

v E LP(fl,Y x Y). We will show that v satisfies a weak Laplace equation.For all 4> E C°°(fl,Y* x Y*) we introduce iß := Q'((ds +L0dt)cß) E C°°(fl,X*).Then

ds(hiß) + dt(J*hiß) = hO'((ds + i*dt)(d8 + hdt)4>)

+ (dsh)iß + dt(hJ*)iß + h(ds@' + J*dtQ')Q'~l(iß).So if iß(s,0) E (J(s,0)T;r(s)/:)± for all (s, 0) G dfl D dM, then hiß is an

admissible test function in the given weak estimate for u in the theorem and

we obtain, denoting all constants by C and using 0*0' = id,

\hv,Acß) (Q(v), h&((-d, + L0dt)(ds + IQdt)4>)

(u,ds(hiß) + dt(rhiß))

; u, (dsh)iß + (dthJ*)iß + h(dsQ' + J*dtQ')Q'~ (iß) )

< (c + ClluHi^n,*))U\\Lp*(n,x*)< C(cu + \\u\\LP(n,x))\\(ß\\w^p*(n,Y*xY*)-

66 CHAPTER 5. CAUCHY-RIEMANN EQUATIONS

Here we used the fact that J* and 0' as well as their first derivatives and in¬

verses are bounded linear operators between Y* x Y* and X*. This inequalitythen holds for all <ß = (fa, fa) with fa E C(fl, Y*) and (ß2 E Cf(fl, Y*) since

in that case iß is admissible. Indeed, iß\t=o = Q'(ds<ßi — dt(ß2,0) G (JT^C)1-due to Q'(Y* x {0}) = &(L0(Y x {O}))1- = (JT^.

Recall that Y C Z is a closed subset of the Banach space Z with the

induced norm. So one also has v E Lp(fl, Z x Z). Let 7r : Z* —y Y* be the

natural embedding, then above inequality holds with (ß = (w o ißx, tt o iß2) for

all ^i EC(fl,Z*) andiß2 ECf(fl,Z*). Since ||7ro^||y. < \\ißt\\z* one then

obtains for all such V = (ipi, iß2) G C°°(ft, Z* x Z*)

< C(cu + ||M||LP(n,x)) ||^||wi.p*(n,z*xz*)-

Now lemma 5.7 (iii) and (iv) asserts the IT^-regularity of hv and hence one

obtains v G W1,p(fl, Z x Z) with the estimate

IMIw^'P^ZxZ) < H^HIiy^^ZxZ) < C(cu + ||«||LP(n,X) + |MUî>(n,ZxZ))-

For the first factor of Z x Z, this follows from lemma 5.7 (iv), in the second

factor one uses (iii). Since it was already known that v takes values in Y x Y

(almost everywhere), one in fact has v E W1,p(fl,Y x Y) with the same

estimate as above. Finally, recall that u = Q o v and use the fact that all

derivatives of O and O-1 are bounded to obtain u E W1,P(K, X) with the

claimed estimate (using again [We, Lemma B.8])

IMIw1^,:*:) < CIHIr^zxz) < C(cu + \\u\\LP(ß^X)).

hv, A^)

U

Chapter 6

Regularity and compactness

This chapter is devoted to the proofs of theorems A and B, restated below as

theorems 6.1 and 6.2. Let (X, r) be a 4-manifold with boundary space-time

splitting. So X is oriented and

X = (J Xk,ken

where all X& are compact submanifolds and deformation retracts of X such

that Xk C int Xfe+i for all k E N. Here the interior of a submanifold X' C

X is to be understood with respect to the relative topology, i.e. we define

int X' = X\ cl(X \ X'). Moreover,

n

dX = \Jrt(StxEt),i=i

where each E8 is a Riemann surface, each St is either an open interval in

M or is equal to S1 = M/Z, and the embeddings t% : S% x E4 —> X have

disjoint images. We then consider the trivial G-bundle over X, where G is a

compact Lie group with Lie algebra g. For i = 1,..., n let £% C *4°'p(£j) be

a Lagrangian submanifold as described in chapter 4, i.e. suppose that

£C.4°£(E,) and ^(E,)%cA.

Furthermore, let X be equipped with a metric g that is compatible with

the space-time splitting r. This means that for each i = 1,... ,n the map

67

68 CHAPTER 6. REGULARITY AND COMPACTNESS

St x [0, oo) x Ej —y X, (s, t, z) \-y 7(s,z)(^) given by the normal geodesies 7(s,z)

starting at 7(s,2)(0) = rz(s, z) restricts to an embedding f1f:lllxHl'^-X for

some neighbourhood Ut C Sz x [0, oo) of St x {0}. Now consider the following

boundary value problem for connections A E A^(X),

(*FA + FA = 0,

\r*A\{s}xEteCt Vse5„î = l,...,n.{')

The anti-self-duality equation is welldefined for A E A^(X) with any p > 1,but in order to be able to state the boundary condition correctly we have

to assume p > 2. Then the trace theorem for Sobolev spaces (e.g. [Ad,Theorem 6.2]) ensures that r;^|{s}xEi G *4°'P(E4) for all s E «S,.

The aim of this chapter is to prove the following two theorems.

Theorem 6.1 Let p > 2. Then for every solution A E A^(X) of (6.1)there exists a gauge transformation u E Q\^(X) such that u*A E A(X) is a

smooth solution.

Theorem 6.2 Letp > 2 and let gu be a sequence of metrics compatible with

t that uniformly converges with all derivatives to a smooth metric. Supposethat Av E A^(X) is a sequence of solutions of (6.1) with respect to the

metrics gv such that for every compact subset K C X there is a uniformbound on \\Fa"\\lp{k)-

Then there exists a subsequence (again denoted Av) and a sequence of

gauge transformations uv E Q\^(X) such that uv*Av converges uniformlywith all derivatives on every compact set to a connection A E A(X).

Both theorems are dealing with the noncompact base manifold X. How¬

ever, we shall use an extension argument by Donaldson and Kronheimer [DK,Lemma 4.4.5] to reduce the problem to compact base manifolds. For the fol¬

lowing special version of this argument a detailed proof can be found in [We,Propositions 8.6,10.8]. At this point, the assumption that the exhausting

compact submanifolds Xk are deformation retracts of X comes in crucially.It ensures that every gauge transformation on Xk can be extended to X,which is a central point in the argument of Donaldson and Kronheimer that

proves the following proposition.

69

Proposition 6.3 Let the 4-manifold M = [Jken Mk be exhausted by compact

submanifolds Mk C int Mk+i that are deformation retracts of M, and let

p>2.

(i) Let A E *4-ioç(M) and suppose that for each k E N there exists a gauge

transformation uk E Q2,p(Mk) such that v*kA\Mk is smooth. Then there

exists a gauge transformation u E G\£(M) such that u*A is smooth.

(n) Let a sequence of connections (Au)ue^ C A^C(M) be given and suppose

that the following holds:

For every k E N and every subsequence of (Al/)pe^ there exist a further

subsequence (vktl)ltEn and gauge transformations uk,t E Q2'p(Mk) such

that

supII^M^H <oo WGN.

Then there exists a subsequence (z^^n and a sequence of gauge trans¬

formations ul E öioc (M) such that

sup \\ul*A^\\ < oo Vfc G N,£ E N.

So in order to prove theorem 6.1 it suffices to find smoothing gauge trans¬

formations on the compact submanifolds Xk in view of proposition 6.3 (i).For that purpose we shall use the so-called local slice theorem. The follow¬

ing version is proven e.g. in [We, Theorem 9.1]. Note that we are dealingwith trivial bundles, so we will be using the product connection as reference

connection in the definition of the Sobolev norms of connections.

Proposition 6.4 (Local Slice Theorem)Let M be a compact 4-manifold, letp > 2, and let q > 4 be such that - > - — \(or q = oo in case p > 4). Fix A E A1,P(M) and let a constant c0 > 0 be

given. Then there exist constants e > 0 and CCG such that the followingholds. For every A E A1,P(M) with

11^4 — A\\q < e and \\A — A\\wi,P < c0

there exists a gauge transformation u E Q2,P(M) such that

( d*^*,4-i) = 0, \\u*A -iH, <CCG\\A-A\\q,s and

\ *(u*A - A)\8M = 0, \\u*A - Â\\wi,P < CCG\\A - A\\wi,P.

70 CHAPTER 6. REGULARITY AND COMPACTNESS

Remark 6.5

(i) If the boundary value problem in proposition 6.4 is satisfied one says

that v*A is in Coulomb gauge relative to A. This is equivalent to v*A

being in Coulomb gauge relative to A for v = u~x, i.e. the boundaryvalue problem can be replaced by

dA(v*Â - A) = 0,

*{v*A-A)\dM = 0.

(ii) The assumptions in proposition 6.4 on p and q guarantee that one has

a compact Sobolev embedding

W1'P(M) m- Lq(M).

(iii) One can find uniform constants for varying metrics in the followingsense. Fix a metric g on M. Then there exist constants e, ô > 0, and

Ccg sucn that the assertion of proposition 6.4 holds for all metrics g'with \\g — g'Hc1 < 5.

In the following we shall briefly outline the proof of theorem 6.1. Given

a solution A E A^C(X) of (6.1) one fixes k E N and proves the assumptionof proposition 6.3 (i) as follows. One finds some sufficiently large compact

submanifold M C X with Xk C M. Then one chooses a smooth connection

Aq e A(M) sufficiently IT^-close to A and applies the local slice theorem

with the reference connection A = A to find a gauge tranformation that

puts A0 into relative Coulomb gauge with respect to A. This is equivalentto finding a gauge transformation that puts A into relative Coulomb gauge

with respect to A0. We denote this gauge transformed connection again byA E A1,P(M). It satisfies the following boundary value problem:

( d*Ao(A-Ao) = 0,

I *FA + FA = 0,

J *(A - Ao)\dM = 0,

( t*A\{s}xSi E £% Vs ESt,i = l,...,n

More precisely, the Lagrangian boundary condition only holds for those

s E St and i E {1,.. .n} for which Tj({s} x E,) is entirely contained in dM.

(6.2)

71

If M was chosen large enough, then the regularity theorem 6.8 below will

assert the smoothness of A on Xk.

The proof outline of the proof of theorem 6.2 goes along similar lines. We

will use proposition 6.3 (ii) to reduce the problem to compact base manifolds.

On these, we shall use the following weak Uhlenbeck compactness theorem

(see [Ul], [We, Theorem 8.1]) to find a subsequence of gauge equivalentconnections that converges LT^-weakly.

Proposition 6.6 (Weak Uhlenbeck Compactness)Let M be a compact 4-manifold and let p > 2. Suppose that the sequence

of connections Av E A1,P(M) is such that ||F4^||P is uniformly bounded.

Then there exists a subsequence (again denoted (A")»^) and a sequence

uv E Ç2,P(M) of gauge transformations such that uu*Au weakly converges

mA1'p(M).

The limit A0 of the convergent subsequence then serves as reference con¬

nection A in the local slice theorem, proposition 6.4, and this way one obtains

a JT^-bounded sequence of connections Av that solve the boundary value

problem (6.2). This makes crucial use of the compact Sobolev embeddingW1,p <—> Lq on compact 4-manifolds (with q from the local slice theorem).The estimates in the subsequent theorem 6.8 then provide the higher Wk,p-

bounds on the connections that will imply the compactness. One difficultyin the proof of this regularity theorem is that due to the global nature of

the boundary conditions one has to consider the E-components of the con¬

nections near the boundary as maps into the Banach space A0,P{T) that

solve a Cauchy-Riemann equation with Lagrangian boundary conditions. In

order to prove a regularity result for such maps one has to straighten out

the Lagrangian submanifold by using coordinates in A0,P(E). This was done

in theorem 5.3 above. Thus on domains U x E at the boundary a crucial

assumption is that the E-components of the connections all lie in one such

coordinate chart, that is one needs the connections to converge strongly in

the L°°(U, Lp(E))-norm. In the case p > 4 this is ensured by the compact

embedding W1>p H L00 on W x E. To treat the case 2 < p < 4 we shall

make use of the following special Sobolev embedding. The proof mainly uses

techniques from [Ad].

Lemma 6.7 Let M,N be compact manifolds and let p > m = dim M and

p > n = dim X. Then the following embedding is compact,

W1'P(M x N) m- L°°(M,LP(N)).

72 CHAPTER 6. REGULARITY AND COMPACTNESS

Proof of lemma 6.7:

Since M is compact it suffices to prove the embedding in (finitely many) co¬

ordinate charts. These can be chosen as either balls B2 C Rm in the interior

or half balls D2 = B2 n Mm in the half space Mm = {x E Rm \ xt > 0} at the

boundary of M. We can choose both of radius 2 but cover M by balls and half

balls of radius 1. So it suffices to consider a bounded set K C W1,P(B2 x N)and prove that it restricts to a precompact set in L°°(Bi, LP(N)), and simi¬

larly with the half balls. Here we use the Euclidean metric on Mm, which is

equivalent to the metric induced from M.

For a bounded subset K, C W1,P(D2 x N) over the half ball define

the subset K' C W1'P(B2 x N) by extending all u E /C to B2 \ Mm by

u(xi, x2,..., xm) := u(—X\,x2,..., xm) for x\ < 0. The thus extended func¬

tion is still IT^-regular with twice the norm of u. So K! also is a bounded

subset, and if this restricts to a precompact set in L°°(Bi, LP(N)), then also

K, C L°°(Di,Lp(N)) is compact. Hence it suffices to consider the interior

case of the full ball.

The claimed embedding is continuos by the standard Sobolev estimates -

check for example in [Ad] that the estimates generalize directly to functions

with values in a Banach space. In fact, one obtains an embedding

W1'P(B2 x N) C W^p(B2,Lp(N)) ^ C°>X(B2,LP(N))

into some Holder space with A = 1 — — > 0. One can also use this Sobolev

estimate for W1,P(N) with A' = 1 — - > 0 combined with the inclusion

LP <^->- L1 on B2 to obtain a continuous embedding

W^P(B2 xN)c Lp(B2,Whp(N)) ^ Lp(B2,C°'x'(N)) C l}{B2,CQ'x'{N)).

Now consider a bounded subset K C W1,P(B2 x N). The first embeddingensures that the functions u E K, u : B2 —y IP(N) are equicontinuous. For

some constant C

\\u(x) - u(y)\\LP(N) < C\x - y\x ^x,yEB2,uEK. (6.3)

The second embedding asserts that for some constant C

u\\c0,x>{N) <C Vue K. (6.4)

In order to prove that K. C L°°(Bi,LP(N)) is precompact we now fix any

e > 0 and show that K. can be covered by finitely many e-balls.

/

73

Let J G C°°(Mm, [0,oo)) be such that supp J C B1 and / J = 1. Then

J$(x) := S~mJ(x/S) are mollifiers for 5 > 0 with supp Js C Z?«5 and J J$ = 1.

Let 5 < 1, then Js * u\Bl E Cca(B1,Lp(N)) is welldefined. Moreover, choose

5 > 0 sufficiently small such that for all u E K,

\Js*u-u\\ = sup / Jö(y) (u(x - y) - u(x)) dmyJbs

< sup [ Js(y)C\y\xdmyleft JBa

LP(N)

<C5X < \e.

Now it suffices to prove the precompactness of /Q := {Js * u | u E /C}, then

this set can be covered by |g:-balls around J$ * u% with u% E K. for i = 1,..., /1 and above estimate shows that K. is covered by the e>balls around the u%.

Indeed, for each u E K one has || J$ * u — J$ * «î||L°o(JB1,LP(iv)) < | ^or some

i = 1,..., / and thus

\\u — Mj|| < 11tt — Js * t/|| + || Js * u — Js * uß\\ + || Js * ut — Mj|| < e.

The precompactness of Ks C L°°(Bi)LP(N)) will follow from the ArzéTa-

Ascoli theorem (see e.g. [L, IX §4]). Firstly, the smoothened functions Js *u

are still equicontinuous on Bt. For all u E K, and x,y E B\ use (6.3) to

obtain

||(«7* * u)(x) - (Js * u)(y)\\LP{N) < / Js(z) \\u(x - z) - u(y - ^)||lp(tv) dJ Bs

< f Js(z)C\x-y\xdmz = C\xJBa

y\BS

Secondly, the L°°-norm of the smoothened functions is bounded by the L1-

norm of the original ones, so for fixed ô > 0 one obtains a uniform bound

from (6.4) : For all m G AC and x E B\

y)\\u(y)\\c^'(N)dmy\\(Js *u)(x)\\co,> '(N)

<c\

Js(x

L\\° 0 11oo

1If a subset K c (X, d) of a metric space is precompact, then for fixed e > 0 one firstlyfinds v\,...,vi G X such that for each x £ K one has d(x,vz) < s for some v%. For each

v, choose one such xl e K, or simply drop v% if this does not exist. Then K is covered

by 2e-balls around the xz: For each x £ K one has d(x,x,) < d(x,v,) + à(vl,x%) for some

i = l,...,J.

74 CHAPTER 6. REGULARITY AND COMPACTNESS

Now the embedding C°'x (N) <^->- LP(N) is a standard compact Sobolev em¬

bedding, so this shows that the subset {(Js * u)(x) \u E /C} C LP(N) is

precompact for all x E B1. Thus the Arzéla-Ascoli theorem asserts that

K-s C L°°(B1,LP(N)) is compact, and this finishes the proof of the lemma.

D

In the proof of theorem 6.2, the weak Uhlenbeck compactness togetherwith the local slice theorem and this lemma will put us in the position to

apply the following crucial regularity theorem that also is the crucial pointin the proof of theorem 6.1. Here (X, r) is a 4-manifold with a boundary

space-time splitting as described in definition 1.1 and in the beginning of this

chapter.

Theorem 6.8 For every compact subset K C X there exists a compact sub¬

manifold M C X such that K C M and the following holds for all p > 2.

(i) Suppose that A E A1,P(M) solves the boundary value problem (6.2).Then A\K E A(K) is smooth.

(ii) Fix a metric g0 that is compatible with r and a smooth connection

Aq E A(M) such that t*^4o|{s}xe, E £% for all s E Sl and i = 1,..., n.

Moreover, fix a compact neighbourhood V = IJlLi ^(^x^) ofKndX.

(Here f0,, denotes the extension ofrt given by the geodesies of g0.) Then

for every given constant C\ there exist constants ô > 0, ôk > 0, and Ck

for all k > 2 such that the following holds:

Fix k > 2 and let g be a metric that is compatible with r and satisfies

\\9 — 9o\\ch+2(M) ^ £>k- Suppose that A E A1,P(M) solves the boundaryvalue problem (6.2) with respect to the metric g and satisfies

11^4 — A||vK1'P(M) < C\,

IK^A-A0)\^\\L^iUiA0,P{Si]) <5 Vi = l,...,n.

Then A\K E A(K) is smooth by (i) and

M — A)||vf*>p(/s:) < Ck.

We first give some preliminary results for the proof of theorem 6.8. The

interior regularity as well as the regularity of the ^-components on a neigh¬bourhood Ui x E, of a boundary component St x Ej will be a consequence

75

of the following regularity result for Yang-Mills connections. The proof is

similar to lemma 3.3 and can be found in detail in [We, Proposition 10.5].Here M is a compact Riemannian manifold with boundary dM and outer

unit normal v. One then deals with two different spaces of test functions,

Cf(M,Q) and C(M,g) as in lemma 3.3.

Proposition 6.9 Let (M, g) be a compact Riemannian 4-manifold. Fix a

smooth reference connection A0 E A(M). Let X E T(TM) be a smooth

vector field that is either perpendicular to the boundary, i.e. X\qM = h v

for some h E C°°(dM), or is tangential, i.e. X\qM E T(TdM). In the firstcase let T = C|°(M,g), in the latter case let T = C£°(M, g). Moreover,let N C dM be an open subset such that X vanishes in a neighbourhood ofdM \ N C M. Let 1 < p < oo and k E N be such that either kp > 4 or k = 1

and 2 < p < 4. In the first case let q := p, in the latter case let q := -^-.Then there exists a constant C such that the following holds.

Let A = A0 + a E Ak,p(M) be a connection. Suppose that it satisfies

dA0a = 0, ((, ^

*«|9M = 0 onNcdM,{ J

and that for all 1-forms ß = <ß txg with <f> E T

[FA,dAß) = 0. (6.6)/Jm' m

Then a(X) E Wk+1>q(M,Q) and

||o;(X)||W/s!+i,„ < C (l + ||o!||Wfc,p + IMI^fc,,,).

Moreover, the constant C can be chosen such that it depends continuously on

the metric g and the vector field X with respect to the Ck+1-topology.

Remark 6.10 In the case k = 1 and 2 < p < 4 the iteration of propo¬

sition 6.9 also allows to obtain IT2'p-regularity and -estimates from initial

ir^-regularity and -estimates.

Indeed, the Sobolev embedding W2,q ^->- W1,p' holds with p' = ^f- since

q < 4. Now as long as p' < 4 one can iterate the proposition and Sobolev

embedding to obtain regularity and estimates in W1,Pl with p0 = p and

4& 2p%P.+i =

-:=

-A> % > Pi-

4 - qt 4-p%

76 CHAPTER 6. REGULARITY AND COMPACTNESS

Since 9 := ^- > 1 this sequence terminates after finitely many steps at some

Pn > 4. Now in case p^ > 4 the proposition even yields W2,PN-regularityand -estimates. In case p^ = 4 one only uses WX'PN for some smaller p'N > |in order to conclude W2,Pn+1 -regularity and -estimates for p'N+1 > 4.

Similarly, in case k = 1 and p = 4 one only needs two steps to reach W2,p

for some p' > 4.

The above proposition and remark can be used on all components of the

connections in theorem 6.8 except for the E-components in small neighbour¬hoods U x E of boundary components S x E. For the regularity of their

higher derivatives in E-direction we shall use the following lemma. The cru¬

cial regularity of the derivatives in direction of U of the E-components will

then follow from chapter 5.

Lemma 6.11 Let k E N0 and 1 < p < oo. Let fl be a compact manifold,let E be a Riemann surface, and equip fl x E with a product metric g^ © g,

where g = (gx)x^n is a smooth family of metrics on E. Then there exists a

constant C such that the following holds:

Suppose that a E Wk,p(fl x E,T*E) such that both d^a and d^a are

of class Wk,p on fl x E. Then V^a also is of class Wk,p and one has the

following estimate on fl x E

||Vsa;||jyfc,p < CdldsöH^/fc.p + Hd^oH^/^ + ||a||w*,p).

Here Vs denotes the family of Levi-Civita connections on E that is given

by the family of metrics g. Moreover, for every fixed family of metrics g

one finds a Ck -neighbourhood of metrics for which this estimate holds with a

uniform constant C.

Proof of lemma 6.11:

We first prove this for k = 0, i.e. suppose that a E LP(fl x E, T*E) and that

ds«, d^a (defined as weak derivatives) are also of class LP. We introduce the

following functions

/ := d^a G Lp(fl x E), g := - *s dsa G Lp(fl x E),

and choose sequences fv, gv GC°°(^xE), and a" G C°°(ftxE,T*E) that con¬

verge to /, g, and a respectively in the Z7-norm. Note that fn f = fng = 0

in LP(YA, so the fv and gv can be chosen such that their mean value over fl

77

also vanishes for all z E E. Then fix z E E and find £", C E C°°(fl x E) such

that

\^(x,z) = 0 VxEfl, \C(x,z)=0 VxEfl.

These solutions are uniquely determined since AE : W^+2'P(E) —y W^P(E) is

a bounded isomorphism for every j E No depending smoothly on the metric,i.e. on x E fl. Here W^f(T,) denotes the space of IT-^-functions with mean

value zero and W|+2'P(E) consists of those functions that vanish at z E E.

Furthermore, let nx : fi1(E) —y /i1(E,^;r) be the projection of the smooth

1-forms to the harmonic part /i1(E) = ker AE = ker danker d^ with respect to

the metric ^ on E. Then 7T is a family of bounded operators from LP(E, T*E)to LTJ'P(E,T*E) for any j E N0, and it depends smoothly on x E fl. So the

harmonic part of àv is also smooth, 7T o àu E C°°(fl x E, T*E). Now consider

av := ds^ + *EdsC + ^ ° ây G C°°(ft xE,T*E).

We will show that the sequence olv of 1-forms converges to a in the Lp-norm

and that moreover Vs»1' is an Lp-Cauchy sequence. For that purpose we will

use the following estimate. For all 1-forms ß E IT1,P(E,T*E) abbreviating

dE = d

WßWw^E) < C(\\d*ß\\LP{s) + ||d/3||Lp(E) + ||7t(/3)||w1,p(s))< C{\\d*ß\\LP(s) + ||d/3||Lp(E) + \\ß\\LP{s)). (6.7)

Here and in the following C denotes any finite constant that is uniform for

all metrics gx on E in a family of metrics that lies in a sufficiently small

(^-neighbourhood of a fixed family of metrics. To prove (6.7) we use the

Hodge decomposition ß = d£ + *d( + 7r(/3). (See e.g. [Wa, Theorem 6.8]and recall that one can identify 2-forms on E with functions via the Hodge* operator.) Here one chooses £, Ç E W2,P{YA such that they solve A£ = d*ßand AC = *dß respectively and concludes from proposition 3.5 for some

uniform constant C

l|d£lki.P(E) < ||Ç||^p(e) < C||d*/3||Lp(E),

ll+dCHw^CE) < IICIIw^e) < C||d/3|Up(S)-

The second step in (6.7) moreover uses the fact that the projection to the

harmonic part is bounded as map 7r : Z7(E,T*E) —> IT1,P(E, T*E).

78 CHAPTER 6. REGULARITY AND COMPACTNESS

Now consider a — av E Lp(fl x E,T*E). For almost all x E fl we have

a(x, ) - au(x, ) E /7(E,T*E) as well as *ds(a(a;, •) - a"(x, )) E LP(Y>) and

d^(a(x, ) — au(x, )) E /^(E). Then for these x E fl one concludes from the

Hodge decomposition that in fact a(x, ) — a"(x, ) E IT1,P(E, T*E). So we

can apply (6.7) and integrate over x E fl to obtain for all v E N

ii nv\PII" " HiP(OxE)

< / \\a{x,-)-av{x,-)\\pLP(Stgx)

<C f {\\dUa - an\\pLV(s) + ||dE(a - a")||^(E) + ||7r(a - ^)||^P(E))v S2

< C(ll/ ~~ / HiP(nxE) + 11^ ~~ 9 lliP(nxE) + If — ä ||Lp(nxS))-

In the last step we again used the continuity of n. This proves the convergence

av —y a in the Lp-norm, and hence Vs«1' —y V^ct in the distributional sense.

Next, we use (6.7) to estimate for all v G N

\LP{Y,,gx)I|Ve«1I^(0xe) = / l|VEa"(z,

<C (lld^^HiP^) + llds^HiP^) + |K||Lp(s)yJn

< C(||dEo: ||LP(0xS) + ||dsci lliP(nxE) + Wa Hlp(îîxe))-

Here one deals with //-convergent sequences d^au = A^y = fv —y f — dsct,— * d^av = A^Çy = gv —y g — — * d-^a, and av —y a. So (Vec^^n is uni¬

formly bounded in LP(fl x E) and hence contains a weakly /^-convergent

subsequence. The limit is V^ct; since this already is the limit in the distri¬

butional sense. Thus we have proven the //-regularity of Vs« on fl x E,and moreover above estimate is preserved under the limit, which proves the

lemma in the case k = 0,

||Vsa||LP(oxE) < liminf HVeck^IIlp^xe)

< liminf C(||dEa,'||Lp(nxE) + HdEölli^nxE) + IK||lj>(qxe))V—»oo

= C(||dsa;||Lp(oxE) + ||dEa|Up(Oxs) + IMIlp(Qxe))-

In the case k > 1 one can now use the previous result to prove the lemma.

Let a G Wk>p(fl x E, T*E) and suppose that ds«, dEct are of class Wkjp. We

79

denote by V the covariant derivative on fl x E. Then we have to show that

V^VsQ; is of class LP. So let Xl5..., Xk be smooth vector fields on fl x E

and introduce

à:=VXl...VXha E Lp(QxE,T*E).

Both ds<5 and dsoi are of class LP since

dsà = [ds, VXl • • • Vxja + VXl • • VXfcdSQ;,

d*sa = [dE, VXl • • • Vxja + VXl • • VXfcd^a.

So the result for k — 0 implies that Vs« is of class LP, hence VfeVs« also is

of class Z7 since for all smooth vector fields

VXl • • • VXk Vs« = [VE, VXl • • • Vxja + VEà.

With the same argument - using coordinate vector fields X% and cuttingthem off - one obtains the estimate

||V VEa||LP(oxs) < C*(||V dEo;||LP(nxs) + ||V dsa||tP(oXE) + ||tt||wM(s)xE))-

Now this proves the lemma,

||Vsa||n't,p(nxE) < II^ECtllw^-i^nxE) + ||V Vsa;||Lp(nxS)< C(||dEa:||Wrfc,p(nxE) + ||dsa||w^p(QxE) + IMIw*.p(î>:xE))-

D

Proof of theorem 6.8 :

Recall that a neighbourhood of the boundary dX is covered by embeddings

fo,î : Wj x E, ^-> X such that Tq^q — ds2 + d£2 + go]S,t- (In the case (i) we

put go :— g ) Since K C X is compact one can cover it by a compact subset

Kint C intX and Khdy := U"=1 f0,,(/0,i X [0, 50] x Ej for some S0 > 0 and

/o,j C »Sj that are either compact intervals in M or equal to S1. Moreover, one

can ensure that -Kbdy C int V lies in the interior of the fixed neighbourhoodof K fl dX. Since X is exhausted by the compact submanifolds Xk one then

finds M :— Xk C X such that both Xbdy and Xint are contained in the inte¬

rior of M (and thus also K C M). Now let ^ G «41,P(M) be a solution of the

boundary value problem (6.2) with respect to a metric g that is compatible

80 CHAPTER 6. REGULARITY AND COMPACTNESS

with r. Then we will prove its regularity and the corresponding estimates in

the interior case on Kint and in the boundary case on Kbdy separately.

Interior case :

Firstly, since Kint c int M and Kint c int X — X \ dX we find a sequence

of compact submanifolds Mk C int X such that Kint C Mfc+1 C int Mk C M

for all k G N. We will prove inductively A\Mh G Ak,p(Mk) for all k G N and

thus A\Kint G A(Kint) is smooth. Moreover, we inductively find constants

Ck, 5k > 0 such that the additional assumptions of (ii) in the theorem imply

M — A)||w/fc.ï>(Mi,) < Ck. (6.8)

Here we use the fixed smooth metric g0 to define the Sobolev norms - for a

sufficiently small (^-neighbourhood of metrics, the Sobolev norms are equiv¬alent with a uniform constant independent of the metric. Moreover, recall

that the reference connection A0 is smooth.

To start the induction we observe that this regularity and estimate are

satisfied for k — 1 by assumption. For the induction step assume this regu¬

larity and estimate to hold for some k G N. Then we will use proposition 6.9

on A\Mk G Ak'p(Mk) to deduce the regularity and estimate on Mk+1.

Every coordinate vector field on Mk+i can be extended to a vector field

X on Mk that vanishes near the boundary dMk. So it suffices to consider

such vector fields, i.e. use N = 0 in the proposition. Then a :— A — A0satisfies the assumption (6.5). For the weak equation (6.6) we calculate for

all ß = cß iXg with (ß G T = C%°(Mk, g)

- f (FA,dAß) = / (dA((ß-ixg)AFA) = [ ((ß-txgAFA) - 0.

Jmu JMh JdMh

We have used Stokes' theorem while approximating A by smooth connections

A, for which the Bianchi identity d^F^ — 0 holds. Now proposition 6.9

and remark 6.10 imply that ^4|mj,+1 £ Ak+1'p(Mk+i). In the case (ii) of the

theorem the proposition moreover provides 5k+i > 0 and a uniform constant

C for all metrics g with \\g — go\\ck+1(Mk) < $k+i sucn that the following holds:

If (6.8) holds for some constant Ck, then

M — A)||w^+i.p(Mj,+1) < C ( 1 + \\A — A)||w^-P(Mj.) + M — ^4o||M/*.P(Mfc)J<C(l + Ck + Cl) =: Ck+1.

81

Here we have used the fact that the Sobolev norm of a 1-form is equivalent to

an expression in terms of the Sobolev norms of its components in the coordi¬

nate charts. In case k — 1 and p < 4, this uniform bound is not found directlybut after finitely many iterations of proposition 6.9 that give estimates on

manifolds Ni = M± and M2 C Xî+1 C int N^ In each step one chooses a

smaller S2 > 0 and a bigger C2. This iteration uses the same Sobolev embed¬

dings as remark 6.10. This proves the induction step on the interior part Amt-

Boundary case :

It remains to prove the regularity and estimates on K\,ây near the boundary.So consider a single boundary component K' := fo(/o x [0, So] x E). We

identify /0 = 51 — M/Z or shift the compact interval such that L0 = [—ro, tq]and hence K' — fo([—r0,r0] x [0, ô0] x E) for some r0 > 0. Since Äbdy (andthus also K') lies in the interior of M as well as V, one then finds /?0 > t0

and A0 > 50 such that f0([-/?0, -Ro] x [0, A0] x E) C M n V. Here f0 is the

embedding that brings the metric go into the standard form ds2 + dt2 + go-s,t-

A different metric g compatible with r defines a different embedding f such

that f*g = ds2 + dt2 + gStt. However, if g is sufficiently C1-close to g0, then

the geodesies are C°-close and hence f is C°-close to fo. (These embeddingsare fixed for t — 0, and for t > 0 given by the normal geodesies.) Thus for a

sufficiently small choice of 52 > 0 one finds R > r > 0 and A > 5 > 0 such

that for all r-compatible metrics g in the 52-ball around g0

K'cf([-r,r] x [0,6] x E) and f([-R, R] x [0, A] x E) C M n V.

(In the case (i) this holds with r0, S0, Ro, and A0 for the fixed metric g — g0.)We will prove the regularity and estimates for f*A on [—r, r] x [0, 5] x E. This

suffices because for Cfc+2-close metrics the embedding f will be Cfc+1-close to

the fixed fo, so that one obtains uniform constants in the estimates between

the VTfc'p-norms of A and f*A. Furthermore, the families gStt of metrics on E

will be Cfc-close to g0.Stt for (s, t) G [—R, R] x [0, A] if 5k is chosen sufficientlysmall. Now choose compact submanifolds flk C H := {(s, t) G M2 | t > 0}such that for all k G N

[-r,r]x[0,5] C flk+1 C int fi* C [-R,R] x [0, A].

We will prove the theorem by establishing the regularity and estimates for

f*A on the flk x E in Sobolev spaces of increasing differentiability. We

distinguish the cases p > 4 and 4 > p > 2. In case p > 4 one uses the

following induction.

82 CHAPTER 6. REGULARITY AND COMPACTNESS

I) Let p > 2 and suppose that A G _4,1,2p(M) solves (6.2). Then we will

prove inductively that fM|nfcxE £ Ak,q(flk x S) for all k G N and with q — p

or q = 2p according to whether k > 2 or k — 1. Moreover, we will find a

constant 5 > 0 and constants Ck,Sk > 0 /or all k > 2 such that the followingholds:

If in addition \\g — go\\ck+2(M) < &k and

\\A — A0\\w12p(m) < Ci,

HTq^ - A0)|e|U°°(W,^Op(E)) < 5,

i/ien /or all k GN

Hf*^ - A))||w*«(fî«.xE) < Cfc-

This is sufficient to conclude the theorem in case p > 4 as follows. One

uses I) with p replaced by \p to obtain regularity and estimates of A — A0

in A1'p(fl1 x E), A2^(fl2 x E), and Ak^(flk x E) for all k > 3. Recall

that the component K' of Äbdy is contained in each f(flk x E). In addition,

one has the Sobolev embeddings Wk+l,ï —>• Wk,p —>• Cfc_1 on the compact

4-manifolds £\+1 x E), cf. [Ad, Theorem 5.4]. So this proves the regularityand estimates on Äbdy-

In the case 4 > p > 2 a preliminary iteration is required in order to

achieve the regularity and estimates that are assumed in I). In contrast to I)the iteration is in p instead of k.

II) Let 4 > p > 2 and suppose that A G A1,P(M) solves (6 2) Then we

will prove inductively that f*^4|njXs E A1,P3(flj x E) for a sequence (p3) with

Pi — p andpJ+i — 9(pJ)-p3, where 9 : (2, 4] —y (1, y|] is monotonely increasing

and thus the sequence terminates with p^ > 4 for some N E N.

Moreover, we will find constants 5 > 0 and constants Ci;J,ôit3 > 0 for

j = 2,..., N such that the following holds:

If for some j = 1,..., N in addition \\g — go\\c3(M) < ^1,3 and

11^4 — ^oIIm^^m) < Ci,

\\fo(A — A))|e||l°°(W,.4°p(E)) < Ö,

then

\\f*(A-Ao)\\wiP}(n z) <Ci,3-

83

Assuming I) and II) we first prove the theorem for the case 4 > p > 2.

After finitely many steps the iteration of II) gives regularity and estimates in

A1<PN(flN x E) with pN > 4 and under the assumption \\g — go||c3(M) < 5i,non the metric. Now if necessary decrease p^ slightly such that 2p > p^ > 4,

then one still has .A^^-regularity and estimates on all components of Äbdyas well as on K-mt (from the previous argument on the interior). Thus the

assumptions of I) are satisfied with p replaced by I^Pn and C\ replaced bya combination of C\^ and a constant from the interior iteration (both of

which only depend on C\). One just has to choose S2 < 5ïjN and choose

the S > 0 in I) smaller than the S > 0 from II). Then the iteration in I)

gives regularity and estimates of A — A0 in Ak'^PN (flk x E) for all k > 2.

This proves the theorem in case 2 < p < 4 due to the Sobolev embeddingsfflk+i^pN ^ y^k,p ^ Ck~2. So it remains to establish I) and II).

Proof of I):The start of the induction k = 1 is true by assumption (after replacing C±

by a larger constant to make up for the effect of f*). For the induction step

assume that the claimed regularity and estimates hold for some k G N and

consider the following decomposition of the connection A and its curvature:

f*A = <S>ds + ^dt + B,

f*FA = FB + (dß$ - dsB) A ds + (dB^ - dtB) A dt (6.9)

+ (<9s*-dt$ + [$,*])dsAdi.

Here $, * G Wk>q(flk x E, fl), and B E Wk'q(flk x E, T*E®fl) is a 2-parameter

family of 1-forms on E. Choose a further compact submanifold fl C int flk

such that flk+\ C intQ. Now we shall use proposition 6.9 to deduce the

higher regularity of $ and \& on fl x E. For this purpose one has to extend

the vector fields ds and dt on fl x E to different vector fields on flk x E, both

denoted by X, and verify the assumptions (6.5) and (6.6) of proposition 6.9.

These extensions will be chosen such that they vanish in a neighbourhood of

(dflk \ dM) x E. Then a := f*(A - A0) satisfies (6.5) on M = f(flk x E)with N = f((dflk n dM) xE).

Choose a cutoff function h E C°°(flk, [0,1]) that equals 1 onO and van¬

ishes in a neighbourhood of dflk \dW. Then firstly, X :— hdt is a vector field

as required that is perpendicular to the boundary dflk x E. For this type of

vector field we have to check the assumption (6.6) for all ß = (ßh dt with

(ß G Cs°(flk x S,fl). Note that f*/3 — (</> • h) o f-1 •

i(f„dt)9 can be trivially

84 CHAPTER 6. REGULARITY AND COMPACTNESS

extended to M and then vanishes when restricted to dM. So we can use

partial integration as in the interior case to obtain

/ (Ff*A,df*Aß) = [ (FA,dAnß) = - [ (nßAFA)Jnhxz Jm JdM

0.

Secondly, X := hds also vanishes in a neighbourhood of (dflk \ dM) x E and

is tangential to the boundary dflk x E. So we have to verify (6.6) for all

ß = (ßh ds with (ß G T — Cß?(flk x E, fl). Again, f*ß extends trivially to M.

Then the partial integration yields

/ (Ff*A,df*Aß) = - f (ßAf*FA)Jnhxi: Jf^idM)

-I {(ßh-dsAFB)=0.J(nkndw)xT,

The last step uses the fact that B(s,0) = r*A|{s}xs E £ C A^t(T,), and

hence FB vanishes on dM x E. However, we have to approximate A bysmooth connections in order that Stokes' theorem holds and FB is well de¬

fined. So this calculation crucially uses the fact that a IT^-connection with

boundary values in the Lagrangian submanifold £ can be IT^-approximatedby smooth connections with boundary values in £c\A(Yß). This was proven in

corollary 4.2. So we have verified the assumptions of proposition 6.9 for both

$ = f*A(ds) and * = f*A(dt) and thus can deduce $, * G Wk+1>q(fl x E).Moreover, under the additional assumptions of (ii) in the theorem we have

the estimates

11$ _ $o||w^+i,«(£}x£) < Cs (l + Ck + Cl) =: Ck+1,

||* - *o|k*+i,«(nxE) < Ct (1 + Ck + Cl) =: Clk+l. (6.10)

The constants Cs and Ct are uniform for all metrics in some small Ck+1-

neighbourhood of go]Stt, so by a possibly smaller choice of 5k+i > 0 theybecome independent of gStt. Note that in the above estimates we also have

decomposed the reference connection in the tubular neighbourhood coordi¬

nates, f*A0 = $o ds + *o dt + B0.

It remains to consider the E-component B in the tubular neighbourhood.

85

The boundary value problem (6.2) becomes in the coordinates (6.9)

Id*Bo(B

- Bo) = Vs($ - $0) + Vf(* - *„),

*FB = dt<S> - ds^ + [*,$],

dsB + *dtB = dB$ + *dß*,^ ' j

S(s,0)g£ V(s,0)GdQfe.

Here dB is the exterior derivative on E that is associated with the con¬

nection B, dB is the coderivative associated with Bo, * is the Hodge op¬

erator on E with respect to the metric gStt, and Va* := 9S$ + [<1>oj$LVt$ :— dt& + [*0?$]- We rewrite the first two equations in (6.11) as a

system of differential equations for a :— B — Bq on E. For each (s, t) G flk

d£a(s, t) = £(s, t), dsa(s, t) - *C(s,i). (6.12)

Here we have abbreviated

f = *[£0 A *(£ - So)] + Vs($ - $o) + Vt(* - *o),

C - - * ds50 -*\[BAB]+ dt<$> - ds* + [*, $].

These are both functions in Wk,q(fl x E, fl) due to the smoothness of A0 and

the previously established regularity of $ and *. (This uses the Sobolev

embedding Wk,q Wk,q °->- ITfc'? due to Wk,q °-> L°°.) So lemma 6.11 asserts

that Vs(-B — Bo) is of class Wk,q on Q x E, and under the assumptions of

(ii) in the theorem we obtain the estimate

||Vs(-B - #o)||wfc>«(nxE)< C(ll£llw*.« + HCllw*.« + \\B - Bo\\Wh,q)< C(l + ll-S — Bo||wfc>« + 11$ ~~ $o||vk*:+1'9 + II* — *o||w*+1>«

+ \\B — Bç,\\Wk,q + ||$ — $o||w*.«||* — *o||vk*.«)< C{1 + Ck + Csk+l + Cl+1 + C2k) =: Cfes+1. (6.13)

Here C denotes any constant that is uniform for all metrics in a Ck+1-

neighbourhood of the fixed go-,s,t, so this might again require a smaller choice

of Sk+i > 0 in order that the constant C]?+1 becomes independent of the

metric gs>t-

Now we have established the regularity and estimate for all derivatives

of B of order k + 1 containing at least one derivative in E-direction. (Note

86 CHAPTER 6. REGULARITY AND COMPACTNESS

that in the case k — 1 we even have /^-regularity with q = 2p where only

/7-regularity was claimed. This additional regularity will be essential for the

following argument.) It remains to consider the pure s- and t- derivatives

of B and establish the /7-regularity and -estimate for VH+1.B on flk+± x E,where Vm is the standard covariant derivative on H with respect to the metric

ds2 + dt2. The reason for this regularity, as we shall show, is the fact that

B G Wk'q(fl, A0,P(T,)) satisfies a Cauchy-Riemann equation with Lagrangian

boundary conditions,

"

dsB + *dtB = G,,

6.14

B(s,0) G £ V(s,0)Edfl.y '

The inhomogeneous term is

G:=dß$ + *dß* g WKq(fl,A°>p(Y)).

Here one uses the fact that Wk'q(fl x £,T*E ® fl) C Wk'q(fl,A°'p(Z)) since

by lemma 5.8 the smooth 1-forms are dense in both spaces and the norm

on the second space is weaker than the IT^-norm on fl x E. Now one has

to apply the regularity theorem 5.3 for the Cauchy-Riemann equation on

the complex Banach space (A0,P(T,), J0). As complex structure J0 we use

the Hodge operator * on E with respect to the fixed family of metrics go]S,t

on E (that varies smoothly with (s,i) G fl). The Lagrangian and hence

totally real submanifold £ C „4°'P(E) is modelled on Z = IT^E^) 0 Rm

(see lemma 4.1 (iii)), which is bounded isomorphic to a closed subspace of

//(E, MP) for some n E N. In the case (ii) of the theorem moreover a familyof connections B0 G C°°(Q,.4.(£)) is given such that Bo(s,0) G £ for all

(s, 0) G dfl and B satisfies

ll-B — B0\\L^(n,A0'P(^)) — \\t*(A — A))|E|U°°(fvt°.p(E))< C\\fo(A-Ao)\x\\l°°(u,a°>p(e)) < CS.

Here one uses the fact that f (O x E) C fo(W x E) lies in a component of

the fixed neighbourhood V of K n dX. The assumption of closeness to Aq

in .A0'P(E) was formulated for f^(A — Ao)|e- However, for a metric g in a

sufficiently small (^-neighbourhood of the fixed metric go the extensions f

and fo are C1-close and one obtains the above estimate with a constant C

independent of the metric. So B G Wk,q(fl, „4°'p(£)) satisfies the assumptionsof theorem 5.3 if 5 > 0 is chosen sufficiently small. (Note that this choice is

87

independent of k E N.) In the case (i) of the theorem one can also choose such

a smooth B0 sufficiently close to B in the L°°(fl, Z7(E))-norm. In order to do

this smooth approximation with boundary values in the Lagrangian, one uses

the Banach submanifold coordinates in lemma 4.1 (iii) as in corollary 4.2.

Now theorem 5.3 asserts B E Wk+1>p(flk+1, A°'P(E)). By lemma 5.8 this

also proves VH+1ß G Lp(flk+1, „4°'P(E)) = IP(flk+1 x £,T*E <g> fl), and this

finishes the induction step f*^4|nfc+1xs £ Ak+l,p(flk+i x E) for the regularitynear the boundary. The induction step for the estimate in case (ii) of the

theorem now follows from the estimate from theorem 5.3,

||V*+1(fl-Bo)||LP(flfc+1xE)< \\B — Bo\\Wk+i,P(nh+1,A0'p(^))< C(l + \\G\\Wk,q^Ao,P^^ + \\B — Bo\\wh^{n,A°'p{^)))< C(\ + Ck + C2 + Ct+l + d+1) =: Cf+1. (6.15)

Here the constant from theorem 5.3 is uniform for a sufficiently small Ck+1-

neighbourhood of complex structures. In this case, these are the families of

Hodge operators on E that depend on the metric gs^. Thus for sufficientlysmall 5k+i > 0 that constant (and also the further Sobolev constants that

come into the estimate) becomes independent of the metric. The final con¬

stant Ck+± then results from all the separate estimates, see the decomposition

(6.9) and the estimates in (6.10), (6.13), and (6.15),

\\f*(A- A0)\\Wk+i,P^h+lxE) < Ck + Csk+1 + Ck+1 + Ck+1 + Ck+1.

Proof of II):Except for the higher differentiability of B in direction of H this iteration

works by the same decomposition and equations as in I). The start of the

induction k = 1 is given by assumption. For the induction step assume that

the claimed IT1,Pk-regularity and -estimates hold for some k G N with pk < 4.

Then proposition 6.9 gives $, * G W2,qk (fl x E) with corresponding estimates

and

3 if pk = 4.

(In the case pk = 4 one applies the proposition only assuming IF^-regularityfor p'k = y

< 4, then one obtains W2,9k-regularity with qk — 3.) Now the

Qk

88 CHAPTER 6. REGULARITY AND COMPACTNESS

right hand sides in (6.12) lie in W1,qh(fl x E), so lemma 6.11 gives Wl,qk-

regularity and -estimates for VSS on fl x E. Next, B G W1'Pk(fl,A°'p(^))satisfies the Cauchy-Riemann equation (6.14) with the inhomogenous term

G g W1'qk(fl x E, T*(fl x E) (8) fl). Now we shall use the Sobolev embedding

W1'qk(fl xE)^ Uk(fl x E) with

Tk

.r24- if Pa < 4,

4qk I 4"P*

4-Çfc12 if pA - 4.

Note that rk > pk > p due to pfc > 2, so that we have G E Lrk(fl, „4°'P(E)).We cannot apply theorem 5.3 directly because that would require the initial

regularity B E W1,2p(fl, .4°'P(E)) for somep > 2. However, we still proceed as

in its proof and introduce the coordinates from lemma 4.1 (iii) that straightenout the Lagrangian submanifold,

Qs,t:Ws,t^A°'p(Z).

Here Ws>t C Y x Y is a neighbourhood of zero, F is a closed subspace of

/7(E,Mm) for some m G N, O is in Cfc+1-dependence on (s,t) in a neigh¬bourhood U C fl of some (so, 0) G fl fl dM and it maps diffeomorphically to

a neighbourhood of the smooth connection O(0) = Bq(so,0). Thus one can

write

B(s,t) = Qs,t(v(s,t)) V(s,t)EU

with v = (v\,v2) G Wl,Pk(U, Y x F). Moreover, we have already seen that

both B and Vs-B are IT^-regular on U x E, so we have the regularityB E W1>qk(U,A1'qh(T,)) C Wl>qk(U,A°>Sk(Y)) with corresponding estimates.

Here we have used the Sobolev embedding IT1,9*(E) ^ LSk(Tß), see [Ad,Theorem 5.4], for

if Pk < |,

sk = { ^e^ if Pk > l

if Pk = 4.

29ft_

2-<7fc

4Pk

8-3pk

iiPk-80

8-Pk

31

2

(Here we have chosen suitable values of sk for later calculations in case pk > |and thus qk > 2.) The special structure of the coordinates Q in lemma 4.1 (iii)

89

(it also is a local diffeomorphism between A°'Sk(T,) and a closed subset of

LSk(E,R2m) since sk > pk > 2) implies that v G Wl«"(U, LSk(E,R2m)),which will be important later on.

The Cauchy-Riemann equation (6.14) now becomes

dsv + Idtv = /,

v2{s,0) = 0 VsGM.

Here / - (d^O)"1 * (d„0) G W^Pk(U, End(F x F)) and

/ = (dvO)-1 (G - dsQ(v) ~ *dtQ(v)) G Uk(U,YxY).

We now approximate / in Uk (U, F x F) by smooth functions that vanish on

dU, then partial integration in (5.5) yields for all <ß G C°°(U, Y* x Y*) and a

smooth cutoff function as in the proof of theorem 5.3

(hv, A^>= [(f,ds(htß)-dt(h-rcß)) + f(F,</>)Ju Ju

[ («i, Wi) + d8(/i<fe)>. (6.16)

u

+

Here F — (Ah)v + 2(dsh)dsv + 2(dth)dtv + h(dtL)dsv - h(dsL)dtv contains

the crucial terms (dtL)(dsv) and (dsI)(dtv) and thus lies in LïPk(U,Y x Y).This is a weak Laplace equation with Dirichlet boundary conditions for hv2,

Neumann boundary conditions for hv±, and with the inhomogenous term in

W~l>rk(U,Y x F). The latter is the dual space of IT^^F* x Y*) with

— + -r= 1. (The inclusion LïPk(JJ) ^-y W~l,Tk(U) is continuous as can

k k

be seen via the dual embedding that is due to^ —r > -H—K;.) Recall^ f-^ Pk/^

that F C Lp(T,,Rm) is a closed subspace. Since rk > p the special regu¬

larity theorem 5.7 for the Laplace equation with values in a Banach space

cannot be applied to deduce hv G W1,Tk(U, Y x Y). However, the general

regularity theory for the Laplace equation extends to functions with values

in a Hilbert space (cf. [We]). So we use the embedding LP(E) ^-y L2(Y>).Then (6.16) is a weak Laplace equation with the inhomogenous term in

ir-1''^,/:2^,^2)) and enables us to deduce hv E Wl>rk(U, L2(E,M2m))and thus v G Wl,Tk(U, L2(E,M2m)) with the corresponding estimates for

some smaller domain U (a finite union of these still covers a neighbourhoodof fl fl dM). Furthermore, recall that v G Wl>qk(U, LSk(T1,M2m)). Now we

90 CHAPTER 6. REGULARITY AND COMPACTNESS

claim that the following inclusion with the corresponding estimates holds for

some suitable pk+i

W^rk(Ü,L2(T)) n Wl>qk(Ü,LSk(Y)) c W1>Pk+1(Ü,LPk+1(Z)). (6.17)

To show (6.17) it suffices to estimate the LPk+1(U x E)-norm of a smooth

function by its Lrk(Ü, L2(E))- and Lqk(U, Z>(E))-norms. Let ct > 2 and

t E [1,2), then the Holder inequality gives for all / G C°°(U x E,

a

La(UxT,)f [ \f\vrJü Jt.

t II f\\Oi—t<I \\J llL2(E)IM II 2%=i, ,

<^ ll f\\t II f\\Oi—t

Lr(U,L>&))\.- .^r^^^p))

Here we abbreviated r :— rk > pk > 2. Now we want

rk(a-t) 2(a-t)qk = —— and sk = (6.18)

Indeed, in the case pk — 4 our choices qk — 3, rk = 12, and Sk = y togetherwith t :— | and a := y

solve these equations. So we obtain pk+\ = a — jj^Pk-In case pk < 4 the first equation gives

4 +1a = pk. (6.19)

8-Ä

If moreover pk > |, then we choose t :— | to obtain ct = 24^g Pa > if-Pfc-This also solves (6.18) with our choice sk — 44^~80, so we obtain pk+i — j^pk.Finally, in case | > pk > 2 one obtains from (6.18)

2

t =

o

P"ô G [1,2).

-pl + 7Pk-8L 7

Inserting this in (6.19) yields ct = 6(pk) Pk with

3^-49(Pk) =

2,7 Q--Pfc + 7pfc

- 8

91

One then checks that 6(2) = 1 and 9'(p) > 0 for p > 2, thus 6(p) > 1 for

p > 2. Moreover, #(|) = |, so 9(p') = y| for some p' G (2,|). Now for

p > p' we extend the function constantly to obtain a monotonely increasingfunction 9 : (2,4] —y (1, -||]. With this modified function we finally choose

Pk+i — 9(pk) -pk for all 2 < pk < 4. This finishes the proof of (6.17) and thus

shows that v G W^Pk+l(Ü, IPk+i(Ej).In addition, note that our choice of pk+± < a will always satisfy pk+± < rk.

In case pk = 4 see the actual numbers, in case pk < 4 this is due to (6.19),t < 2, and pk > 2,

6 2öl < Pk <

~. Pk = rk.8-pk 4-pk

Now we again use the special structure of the coordinates O in lemma 4.1 (iii)to deduce that B = Q o v E W^Pk+l(lJ',A°'Pk+1(T,)) with the correspond¬

ing estimates. Above, we already established the IT1,r*- and thus TT1'P*+1-

regularity and -estimates for $ and * as well as B G LPk+1(U,A1,Pk+1(Y,)).(Recall the Sobolev embedding Wl'qk —>• Uk, and that pk > qk and rk > pk+\,

so we have Uk{JJ, Lrfc(E))-regularity of B as well as Vs-S.) Putting all this

together we have established the Wl,Pk+1 -regularity and -estimates for f*A

over Ul x E, where the U% cover a neighbourhood of flk^\ fl dM. The interior

regularity again follows directly from proposition 6.9.

This iteration gives a sequence (pk) with pk+x — 9(pk) Pk > 9(p) pk. So

this sequence grows at a rate greater or equal to 9(p) > 9(2) — 1 and hence

reaches p^ > 4 after finitely many steps. This finishes the proof of II) and

the theorem. D

Proof of theorem 6.1 :

Fix a solution A G A^(X) of (6.1) with p > 2. We have to find a gauge

transformation u G G^P (X) such that u*A E ^l(X) is smooth. Recall that the

manifold X = UfceN^fc ^s exhausted by compact submanifolds Xk meetingthe assumptions of proposition 6.3. So it suffices to prove for every k E N

that there exists a gauge transformation u G Q2,p(Xk) such that v*A\Xk is

smooth.

For that purpose fix k G N and choose a compact submanifold M c X

that is large enough such that theorem 6.8 applies to the compact subset

K := Xk c M. Next, choose A0 E A(M) such that \\A - A0\\wi,P(M)and \\A — A)||l«(m) are sufficiently small for the local slice theorem, propo¬

sition 6.4, to apply to AQ with the reference connection A = A. Here due

92 CHAPTER 6. REGULARITY AND COMPACTNESS

to p > 2 one can choose q > 4 in the local slice theorem such that the

Sobolev embedding W1,P(M) <—y Lq(M) holds. Then by proposition 6.4 and

remark 6.5 (i) one obtains a gauge transformation u E Q2,P(M) such that u*A

is in relative Coulomb gauge with respect to A0. Moreover, u*A also solves

(6.1) since both the anti-self-duality equation and the Lagrangian submani¬

folds £l are gauge invariant. The latter is due to the fact that u restricts to a

gauge transformation in Q1,p(Et) on each boundary slice rt({s} x Ej due to

the Sobolev embedding Ç2'P(U, x E) C Wl'p(U%, Ql'p(T,%)) ^ C°(Ut, Gl'p(E%)).So u*A G A1,P(M) is a solution of (6.2) and theorem 6.8 (i) asserts that

u*A\Xk E A(Xk) is indeed smooth.

Such a gauge transformation u G Q2'p(Xk) can be found for every k G N,hence proposition 6.3 (i) asserts that there exists a gauge transformation

u G ^io'c(X) on the full noncompact manifold such that u*A G A(X) is

smooth as claimed.

Proof of theorem 6.2 :

Fix a smoothly convergent sequence of metrics gv —y g that are compatibleto r and let Au G A^X) be a sequence of solutions of (6.1) with respect

to the metrics gv. Recall that the manifold X = \Jke^Xk is exhausted

by compact submanifolds Xk meeting the assumptions of proposition 6.3.

We will find a subsequence (again denoted Av) and a sequence of gauge

transformations uv G Q\^C(X) such that the sequence uv*Av is bounded in

the IT£'p-norm on Xk for all l G N and k G N. Then due to the compact

Sobolev embeddings Wl,p(Xk) <—y Cl~2(Xk) one finds a further (diagonal)subsequence that converges uniformly with all derivatives on every compact

subset of X.

By proposition 6.3 (ii) it suffices to construct the gauge transformations

and establish the claimed uniform bounds over Xk for all k G N and for

any subsequence of the connections (again denoted Av). So fix k G N and

choose a compact submanifold M C X such that theorem 6.8 holds with

K = Xk C M. Choose a further compact submanifold M' C X such that

theorem 6.8 holds with K = M C M'. Then by assumption of the theo¬

rem ||F4^ 11£p(m') is uniformly bounded. So the weak Uhlenbeck compactness

theorem, proposition 6.6, provides a subsequence (still denoted A"), a limit

connection A0 E Al,p(M'), and gauge transformations uv G Q2,P(M') such

that uv *AV —y A0 in the weak IF^-topology. The limit A0 then satisfies

the boundary value problem (6.1) with respect to the limit metric g. So as

93

in the proof of theorem 6.1 one finds a gauge transformation uq G Q2'P(M)such that UqA0 G A(M) is smooth. Now replace A0 by UqA0 and uv by

vPuo G Ç2'P(M), then one still has a IT^-bound, \\uv*Av - A0\\wi,P(M) < c0

for some constant Co, see lemma A.5.

Due top > 2 one can now choose q > 4 in the local slice theorem such that

the Sobolev embedding Wl,p(M) ^-y Lq(M) is compact. Hence for a further

subsequence of the connections uu*Av —y Aq in the Lq-norm. Let e > 0

be the constant from proposition 6.4 for the reference connection A = Aq,then one finds a further subsequence such that ||u1/*^41/ — >Ao||l«(m) < £ for

all veN. So the local slice theorem provides further gauge transformations

v7 E Q2'P(M) such that the vP *AV are in relative Coulomb gauge with respect

to A0. The gauge transformed connections still solve (6.1), hence the vP*Av

are solutions of (6.2). Moreover, we have H^*^—Ao\\q < CCG\\uu*Au—AQ\\q,hence vP *AU —y A0 in the L9-norm, and

\\ü»*A»-Ao\\whp{M)<CCGco.The higher ITfc'p-bounds will now follow from theorem 6.8, so we first have to

verify its assumptions. Fix the metric go := g and a compact neighbourhoodV - Ur=i fo,t (U* x E0 of K n dX- Then the % («" *A" ~ Ao) |e, are uniformlyJT^-bounded and converge to zero in the L9-norm on U% x S% as seen above.

Now the embedding

W^p(Ut x E„ T*E, ® fl) ^ L°°(UU A°'P(K))

is compact by lemma 6.7. Thus one finds a subsequence such that the

fott(üu*Av)\^i converge in L°°(l(t,A0'p(Zt)). The limit can only be f0*^0|Slsince this already is the L?-limit. Now in theorem 6.8 (ii) choose the constant

Ci — CCGco and let 5 > 0 be the constant determined from C\. Then one

can take a subsequence such that

\\fo,%{üv*A" ~ ^o)e,||l°°(u,,.a°.i>(e,)) ^S Vi = l,...,n,Vu.

Now theorem 6.8 (ii) provides the claimed uniform bounds as follows. Fix

l G N, then H^" — g\\ct+2(M) < àe f°r all v>vt with some vi G N, and thus

¥U*AU-A4w^{xk)<ct v»>^

This finally implies the uniform bound for this subsequence,

supH^M"!!wl,p(x) <oo.

94 CHAPTER 6. REGULARITY AND COMPACTNESS

Here the gauge transformations vP G Ç2'p(Xk) still depend on k E N and are

only defined on Xk. But now proposition 6.3 (ii) provides a subsequence of

(Av) and gauge transformations uv E Q\^.(X) defined on the full noncom¬

pact manifold such that vP *AU is uniformly bounded in every IT^'p-norm on

every compact submanifold Xk. Now one can iteratively use the compact

Sobolev embeddings Wi+2>p(Xi) ^ Cl(XA for each £ E N to find a further

subsequence of the connections that converges in Cl(XA. If in each step one

fixes one further element of the sequence, then this iteration finally yieldsa sequence of connections that converges uniformly with all derivatives on

every compact subset of X to a smooth connection A E A(X).

Chapter 7

Fredholm theory

This chapter concerns the linearization of the boundary value problem (1.2)in the special case of a compact 4-manifold of the form X = S1 x Y, where

F is a compact orientable 3-manifold whose boundary dY = E is a disjointunion of connected Riemann surfaces. The aim of this chapter is to prove

theorem C. The parts (i), (ii), and (iii) of theorem C are restated and proven

below as theorem 7.1, lemma 7.2, and lemma 7.3.

So we equip S1 x Y with a product metric g — ds2 + gs (where gs is

an S^-family of metrics on F) and assume that this is compatible with the

natural space-time splitting of the boundary dX = S1 x E. This means that

for some A > 0 there exists an embedding

t : S1 x [0, A) x E m- S1 x Y

preserving the boundary, r(s, 0, z) = (s, z) for all s G S1 and z G E, such

that

r*g = ds2 + dt2 + gs,t.

He-re gs,t is a smooth family of metrics on E. This assumption on the

metric implies that the normal geodesies are independent of s G S1 in

a neighbourhood of the boundary. So in fact, the embedding is given by

r(s,t, z) — (s,7;j(t)), where 7 is the normal geodesic starting at z E E. This

seems like a very restrictive assumption, but it suffices for our applicationto Riemannian 4-manifolds with a boundary space-time splitting. Indeed,the neighbourhoods of the compact boundary components are isometric to

S1 x F with F = [0, A] x E and a metric ds2 + dt2 + gSyt.

Now fix p > 2 and let £ C ,4.flft(E) be a gauge invariant Lagrangiansubmanifold of .4°'P(E) as in chapter 4. Then for A G A1,P(S1 x Y) we

95

96 CHAPTER 7. FREDHOLM THEORY

consider the boundary value problem

- n

(7.1)*FÂ + FÂ = 0,

Ä\{s}xdY G £ Vs G S1.

Fix a smooth connection A G A(S1 x Y) with Lagrangian boundary val¬

ues (but not necessarily a solution of this boundary value problem). It

can be decomposed as A — A + $ds with $ G C°° (S1 x Y, fl) and with

A E C0O(5'1 x F,T*F <8)fl) satisfying As := A(s)\dY G £ for all s G S1. Sim¬

ilarly, a tangent vector à to A1,P(S1 x F) decomposes as à — a + tpds with

<p E W1*^1 x F,fl) and a E W1'P(S1 x Y,T*Y <g> fl). With respect to this

splitting the linearization of the Lagrangian boundary condition in (7.1) at

A is a(s)\oY G Tas£ for all s G S1. Moreover, the linearization of the

anti-self-duality equation FP = \(F^ + *F^) = 0 at A can be expressed as

d^ct = | * (Vsa — dAip + *d^ct) — \ (Vsct — dAip + *d^ct) A ds = 0.

Here dA denotes the exterior derivative corresponding to the connection A(s)on F for all s G S1, * denotes the Hodge operator on F with respect to the

s-dependent metric gs on F, and we use the notation Vsa := dsa + [$, a].Now the linearized operator for the boundary value problem (7.1) has to be

augmented with the local slice condition at A, i.e. the condition that the

tangent vectors à to the space of connections at A lie in a complement of

the gauge orbit through A. This complement is fixed by the Coulomb gauge

conditions

d^ci —

—Vsy + d^ct = 0 and ^àt\s\XQY —

— *ol\qy Ads = 0.

Now let E1/ C IT1'^1 x F,T*F <g> g) be the subspace of ^-families of 1-

forms a that satisfy the above boundary conditions from the linearization of

(7.1) and the Coulomb gauge,

*ct(s)|ay = 0 and a(s)\ßYETAs£ for alls G S1.

Then the linearized operator for the study of the moduli space of gauge

equivalence classes of solutions of (7.1) is

£>(A,*) : F1/ x W^p(Sl x F,fl) — Lp(Sl x F,T*F ® fl) x Lp(Sl x F,0)

given by

D{A^)(a, ip) = (Vsa - dA(p + *dAa, Vsip-d*Aa).

97

Our main result, theorem C (i), is the following.

Theorem 7.1 Let p > 2 and assume that A + Qds G A(Sl x F) satisfies the

boundary condition A(s)\qy E £ for all s E S1. Then D(A^ is a Fredholm

operator.

We now give an outline of the proof of this theorem. The first crucial

point of this Fredholm theory is the following estimate, theorem C (ii), which

ensures that F>(a,$) has a closed image and a finite dimensional kernel.

Lemma 7.2 There is a constant C such that for all à E IT1,P(X, T*X ® fl)satisfying

*ä\dx = 0 and ä\{s}xdY £ TAs£ Vs G S*1

one has the estimate

\\ä\\wi,P < C(||d^ä||p + ||d^ct||p + ||ct||p).

Postponing the proof we first note that by the above calculations the

estimate in this lemma is equivalent to the following estimate for all a E EA,Pand <p E W1'P(S1 xF,ß)

||(ct, ^>)||vpi,P = ||ct||jyi,p + ||</?||wi,p

< C(||Vsa - dAip + *dAa\\p + \\Vs<p - d*Aa\\p + \\a\\p + \\(p\\p)= C(\\DiA>*){a,<p)\\p+\\{a,<p)\\p).

The second part of the proof of theorem 7.1 is the identification of the

cokernel of the operator with the kernel of a slightly modified linearized

operator, which will be used to prove that the cokernel is finite dimensional.

To be more precise let a : S1 x Y —y S1 x Y denote the reflection given by

a(s,y) := (—s,y), where S1 = M/Z. Then we will establish the following

duality:

(ß,C)e{un.D(A,*))± ^^ {ßoa,Coa) G kerFv^,*),

where -Do-*(a,*) is the linearized operator at the connection a*A = Aoa— $ocr

with respect to the metric a*g on S1 x Y. Once we know that im D(a,$) is

closed, this gives an isomorphism between (coker D(a,*))* — (imD(^))1 and

98 CHAPTER 7. FREDHOLM THEORY

kerDa*(^4$). Here Z* denotes the dual space of a Banach space Z, and for

a subspace F C Z we denote by YL C Z* the space of linear functionals

that vanish on F. Now the estimate in lemma 7.2 will also apply to Da*^A^,and this implies that its kernel - and hence the cokernel of D^A^ - is of

finite dimension. The main difficulty in establishing the above duality is

the following regularity result, theorem C (iii). As before we shall use the

notation - + 4r = 1.p p*

Lemma 7.3 Let q > p* such that q ^ 2. Let ß E Lq(S1 x Y,T*Y (g> g),C G Lq(S1 x Y,q), and suppose that there exists a constant C such that forall a G E1/ and ip E W1^P(S1 x F, g)

(D(A)*)(a,^),(/3,C)> < C\\(a,V)\\q*.SlxY

Then ß E W1'"^ x F, T*F ® fl) and ( E W^iS1 x F, fl) .

This regularity as well as the estimate in lemma 7.2 will be proven anal¬

ogously to the nonlinear regularity and estimates in chapter 6. Again, the

interior regularity and estimate is standard elliptic theory, and one has to

use a splitting near the boundary. We shall show that the S1- and the nor¬

mal component both satisfy a Laplace equation with Neumann and Dirichlet

boundary conditions respectively. The E-component will again gives rise to

a (weak) Cauchy-Riemann equation in a Banach space, only this time the

boundary values will lie in the tangent space of the Lagrangian. In contrast

to the required ZAestimates we shall first show that the L2-estimate for LP-

regular 1-forms can be obtained by more elementary methods. These were

already outlined in [Sal] as a first indication for the Fredholm property of

the boundary value problem (7.1).Let et G IT1,P(X, T*X ®fl) be as supposed for some p > 2. From the first

boundary condition *à\ox = 0 one obtains

||Và||2 = ||dà||2 + ||d*â||2- / g(Ya,VY&u).Jax

Here one has Jdxg(Yä, Vys^) > —^11^111,2(9^) since the vector field Fâ is

given by iy&9 = à. In this last term one uses the following estimate for

general 1 < p < oo.

99

Let r : [0, A) x dX —> X be a diffeomorphism to a tubular neighbourhoodof dX in X. Then for all 5 > 0 one finds a constant C$ such that for all

/ G W1'P(X)

LP{dX)1d

ax Jo ds-((S-l)\f(T(s,z))\P)dsd3Z

<[ f \f(r(s,z))\pdsd3z+ f [ p|/(r(s,z))r1|9J(r(s,^))|dsd3^Jax Jo Jax Jo

<C{\WLP{x)^\\f\V{Xpf\\LP{X))<W\\w^p{x)+Cs\\f\\LP{x))v. (7.2)

This uses the fact that for all x, y > 0 and 5 > 0

„_i ^1 5PVP ;a;<55T:T'y I

„ /, ,l_ xp

£p 1/ < S p

~

p } < (5y + S p-ix) .

\ ö~p^xp ;x > ôp^y J~ V y

So we obtain

INIw/1,2 < C(||dj[â||2 + ||d^ct||2 + ||ci;||2). (7.3)

In fact, the analogous IT^-estimates hold true for general p, as is proven e.g.

in [We, Theorem 6.1]. However, in the case p = 2 one can calculate further

for all S > 0

lld«ll2 = / {dÀà, 2djä>- / (d^àAd^à)Jx Jx

= 2\\d±ä\\l- {äA[FÄAä])- (öAd^ä)Jx Jax

< 2||dtà||2 + C«?||ct||2 + <J||â||^M. (7.4)

Here the boundary term above is estimated as follows. We use the universal

covering of S1 = R/Z to integrate over [0,1] x dY instead of dX = S1 x dY.

Introduce A := (As)seSi, which is a smooth path in £. Then using the

splitting ä\ßX = ct + <pds with a : S1 x E —y T*E ® fl and <p : S1 x E —y fl

100 CHAPTER 7. FREDHOLM THEORY

one obtains

— / ( a. A d^à )Jax

= — / ( if , d^ct ) dvols A ds — / / ( ct A (dAcp — Vsct) ) A ds

Jo Jt Jo Jt

= / (et A Vsct) AdsJo Jt.

< 5llâll^i.2(x) +C,5||ct|||2(x).

Firstly, we have used the fact that d^ctls = 0 since a(s) E TAs£ C kerd^sfor all s G S1. Secondly, we have also used that both a and dAip lie in TA£,hence the symplectic form Js(o; A dAtp) vanishes for all s G S1. This is not

strictly true since ex only restricts to an F^-regular 1-form on dX. However,as 1-form on [0,1] x F it can be approximated as follows by smooth 1-forms

that meet the Lagrangian boundary condition on [0,1] x E.

We use the linearization of the coordinates in lemma 4.1 (iii) at As for all

s G [0,1]. Since the path s-ï As E £ fl A(T,) is smooth, this gives a smooth

path of diffeomorphisms Qs for any q > 2,

ZxZ —-* F«(E,T*E®0)W*:

(C,v,C,w) .— d>U + Ei«,7.(s) + *d,i.C + Ei«',*7.(s),

where Z := M^>«(E,g) x Rm and the 7, G C°°([0,1] x E,T*E <g> g) satisfy

7î(s) G T^s£ for all s G [0,1]. We perform above estimate on [0,1] x F

since we can not necessarily achieve 7,(0) = 7«(1). In these coordinates, we

mollify to obtain the required smooth approximations of à near the bound¬

ary. Furthermore, we use these coordinates for q = 3 to write the smooth

approximations on the boundary as a(s) = dAaÇ(s) + Ei vl{s)lz{s) with

||£(s)||wi.3(E) + K*)| < C||ct(s)||L3(E). Then for all s G [0,1]

/"(ct(s)AVs«(s))= f(aA(dAM + Y,T=idsV*-^))Je Jt,

+ J (a A ([$,a] + [cVU] + £I^ 9-7.) >

<C||a(s)||Li(E)||a(s)||L3(E).

Here the crucial point is that dAds£, and dsvt-'yl are tangent to the Lagrangian,hence the first term vanishes. Now one uses (7.2) for p = 2 and the Sobolev

101

trace theorem (the restriction IT1,2(X) —y L3(dX) is continuous by e.g. [Ad,Theorem 6.2] ) to obtain the estimate,

flJo Jt

(aAVsa) Ads < C\\à\\L2{dx)\\à\\L3{dx)T

< ^W^Ww-^ix) + C(5||ö:||L2(x)||ä||^i,2(x)< 5l|à||^M(x) + C^||à|||2(x).

This proves (7.4). Now 5 > 0 can be chosen arbitrarily small, so the term

11 à 11 ^1,2 can be absorbed into the left hand side of (7.3), and thus one obtains

the claimed estimate

||ä||jyi,2 < C(||d^ct||2 + ||d^o:||2 + ||ct||2).

Proof of lemma 7.2 :

We will use lemma 3.3 for the manifold M := S1 x Y in several different

cases to obtain the estimate for different components of à. The first weak

equation in lemma 3.3 is the same in all cases. For all r\ E C°°(M; g)

Jm(à,dr/)= (d*ä,r])+ (rj, *ä)

Jm JdM

= [ (d*Ää + *[ÄA*ä],V) = f (f,V).Jm Jm

Here one uses the fact that *ct|dM = 0.Then / G LP (M, g) and

imiP<||d^â||p + 2||I|U|ct||p. (7.5)

For the second weak equation lemma 3.3 we obtain for all A G fi1 (M; g)

(ct,d*dA) = / (ct,d*dA + d**dÀ) (7.6)m Jm

= / (7,dA)- / (âA*dA)- / (ciAdA),Jm Js1xôy JslxdY

where 7 = da + *dct = 2dtâ - 2[Ä A à}+ G IP(M, A2T*M <g> fl) and

||7||p<2||d+à||p + 4||i|U|ct||p. (7.7)

Now recall that there is an embedding r : S1 x [0, A) x E <^-y S1 x Y to a

tubular neighbourhood of 5*1 x dY such that r*g = ds2 + dt2-\-gSjt for a family

102 CHAPTER 7. FREDHOLM THEORY

gs,t °f metrics on E. One can then cover M = S1 x Y with r(S1 x [0, ^] x E)and a compact subset V C M \ dM.

For the claimed estimate of ct over V it suffices to use lemma 3.3 for vector

fields X G r(TM) that equal to coordinate vector fields on V and vanish on

dM. So one has to consider (7.6) for A = (ß iXg with <ß E Cf(M,g). Then

both boundary terms vanish and hence lemma 3.3 directly asserts, with some

constants C and Cy, that

||ct||M/i.p(y) < C(||/||lp(m) + HtIUp(m) + IH|lp(m))< CV(l|d|ct||Lp(M) + ||d^o:||Lp(M) + ||o:||lp(m))-

So it remains to prove the estimate for à near the boundary dM = S1 x E.

For that purpose we can use the decomposition r*ä = <pds + ißdt + a, where

<p,ipe W^P(SX x [0, A) x E, ß) and a E W1*^ x [0, A) x E, T*E <g> g). Let

fl ;= S1 x [0, |A] and let K := S1 x [0, f ]. Then we will prove the estimate

for ip and iß on fl x E and for et on K x E.

Firstly, note that iß = ä(r*dt) o r, where —T*dt\dM = v is the outer unit

normal to dM. So one can cut off r*dt outside of r(fl x E) to obtain a

vector field X G T(TM) that satisfies the assumption of lemma 3.3, that is

X\dM = —v is perpendicular to the boundary. Then one has to test (7.6)with A = <f> txg for all (ß E Cg°(M,g). Again both boundary terms vanish.

Indeed, on 5*1 x dY we have <f> = 0 and txg = r*dt, hence dA^^y = 0

and *dA|Kx9Y = —^ * T"*(dt A dt) = 0. Thus lemma 3.3 yields the followingestimate.

H^llwLptnxE) < C\\ä(X)\\wi,P{M)< C(||/||lp(M) + ||7|Up(M) + ||ä||LP(Af))< Ct(||d^ä||Lp(Af) + ||d^ct||Lp(M) + ||ct||Lp(M)).

Here C denotes any finite constant and the bounds on the derivatives of r

enter into the constant Ct.

Next, for the regularity of ip = à(ds) o r one can apply lemma 3.3 with

the tangential vector field X = ds. Recall that r preserves the S^-coordinate.

One has to verify the second weak equation for all (ß E C(M, g), i.e. consider

(7.6) for A = cß ixg = 4> " ds. The first boundary term vanishes since one has

*d\\siXdY = — ye- dvobjy = 0. For the second term one can choose any S > 0

103

and then finds a constant Cs such that for all <f> G C^(M, g)

à AdA"S1xdY

S1xdY

ct(s,0) AdE(</>or)(s,0)) Ads

( & A [Ä, cß] ) A ds

< Hal LP(dM) LP*(dM)

< (5||ä||iyi,p(Af) + Q||ö||lp(m)) tyi,p*(M)-

This uses the fact that ct(s, 0) and dAs((ß o r)|(S]0)XE both lie in the tangent

space TAa£ to the Lagrangian, on which the symplectic form vanishes, that is

Js( aAdA((ßor) ) = 0. Moreover, we have used the trace theorem for Sobolev

spaces, in particular the estimate (7.2). Now lemma 3.3 and remark 3.4 yieldwith a = \\f\\p, c2 = ||7||lp(m) +5||ä||wMM) + Cs\\à\\LPiM), and using (7.5),(7.7)

WfWw-^inxT)< C{\\f\\LP{M) + C2 + ||ct||Lp(M))< 5||à||M/i,p(M) + Cs(5)(||d^Ct||Lp(M) + ||d^Ct||Lp(M) + ||ci||lp(m))-

Here again S > 0 can be chosen arbitrarily small and the constant CS(S)depends on this choice.

It remains to establish the estimate for the E-component a near the

boundary. In the coordinates r on fl x E, the forms d*rct and d^ct become

r*d*Ää = -ds<p - dtiß + dE« - T*(*[Ä A *«]),

T*dß~ä = |(—(<9sct + *sdta) A ds + *s(dsa + *sdta) A dt)+ i(dEct+ (*EdEct)ds A dt) + r*([I A ct]+).

So one obtains the following bounds: The components in the mixed direction

of fl and E of the second equation yields for some constant C\

||dsa + *EcV*||Lp(nxE) < lk*dtà||LnnxE) + ||r*([iAQ;]+)||LnnxS)< CiOld^ctH^M) + ||ö||lp(m))-

Similarly, a combination of the first equation and the E-component of the

second equation can be used for every 5 > 0 to find a constant C2(S) such

104 CHAPTER 7. FREDHOLM THEORY

that

||dECt||Lp(nXE) + ||dECl||Lp(nXE)< C(||dtct||Lp(M) + ||d^ct||Lp(M) + ||ä||LP(Af)

+ |M|w^î>(nxE) + II^II^LPCnxE))< 5||ä||jyi,p(M) + C2(5)(||d^ct||Lp(M) + ||d^ct||Lp(M) + ||ci||m))-

Now firstly, lemma 6.11 provides an LP-estimate for the derivatives of a in

E-direction,

||VEct||Lp(nxE)< C(||dEci||Lp(nxs) + ||dEct||Lp(nxE) + |H|Lp(fiXE))< 5||ct||H/i,p(M) +C,E(5)(||d^Cl||Lp(M) + ||d*|Ct||Lp(M) + ||o;||lp(m)),

where again CE(5) depends on the choice of S > 0. For the derivatives

in s- and ^-direction, we will now apply theorem 5.4 on the Banach space

X = LP(T,, T*E®g) with the complex structure *E determined by the metric

gSjt on E and hence depending smoothly on (s, t) E fl. The Lagrangianand hence totally real submanifold £ C X is modelled on a Banach space

Z = W^ÇE, fl) ©Mm as seen in lemma 4.1, and this is bounded isomorphic to

a closed subspace of LP(E, Rn). Now a G W1,p(fl, X) satisfies the Lagrangian

boundary condition a(s, 0) G TAs£ for all s G S1, where s-+ As is a smooth

loop in £. Thus corollary 5.5 yields a constant C such that the followingestimate holds,

||VnCt||LP(^xE) < ||ct||vKi.p(K,x)< C(||<9sct + *zdta\\LP(n;X) + |H|Lp(n,x))< C/<:(||d^à||Lp(M) + ||<S||lp(m))-

Here Ck also includes the above constant C\. Now adding up all the esti¬

mates for the different components of à gives for all 5 > 0

||ct||wn,P < (Cv + Ct + CS(S) + CE(5) + CW)(||dtä||p + ||d^â||p + ||ct||p)+ 25||ct||wn,p.

Finally, choose ô =x

j,then the term ||ct||tyi,p can be absorbed into the left

hand side and this finishes the proof of the lemma.

105

Proof of lemma 7.3 :

Let ß E Lq(S1 x F,T*F (g> ß) and C e ^(S1 x Y,g) be as supposed in

the lemma. Then there exists a constant C such that for all et G EA,P and

cpEW1'p(S1 xF,ß)

Js1 Jy( Vsa - dA<p + *dAa, ß ) +

(D{A<s>)(a,ip), (ß,0)'S1xY

<C\\(a,V)\\q*.

II'Js1 Jy

Vs<p-d*Aa, C)

(7.8)

The higher regularity of (, is most easily seen if we go back to the notation

ct = ct + (pds with D(a,*) (a, <p) = (2j, —d*~ct) for d~tct = *7 —

7 A ds.

Abbreviate M := S1 x Y, then we have for all à G C°°(M,T*M <g> g) with

*ct|aM = 0 and à\{s}xdY £ ^as£ for all s G 5*1

/ (2d+ä,ßAds) + [ (d*Ää,C)Jm Jm

<C\\ä\ q*-

Now use the embedding r : S1 x [0, A) x E °->- M to construct a connection

 G ^4(M) such that t*Â(s, t, z) = As(z) near the boundary (this can be cut

off and then extends trivially to all of M). Then à := dA(ß satisfies the above

boundary conditions for all (ß E C(M, g) since dA(ß(v) = f£ + [A(v), <f>] = 0

and d^(ß\{syXdY = d^^» G TAs£ for all s E S1. Thus we obtain for all

(ß E C^(M, fl) in view of A(ß = d*(ct — [A, <ß]), denoting all constants by C,

JM

A^,C)

(d^ct + *[iA*à]-d*[i>],C)M

<C||ä||,.+

< CMWw

JM

2d\dA(ß,ßAds) + (*[AA*dÂ<f>]-d*[Â,<l>],Ç)M

The regularity theory for the Neumann problem, e.g. proposition 3.5, then

asserts that Ç E W1'q(M).

106 CHAPTER 7. FREDHOLM THEORY

To deduce the higher regularity of ß we will mainly use lemma 3.3. The

first weak equation in the lemma is given by choosing ct = 0 in (7.8). For all

nEC°°(M,g)

(ß,dn]M

f f(ß,dAri-[A,Ti])Js1 Jy

Js1 Jy<C|H|,.+

<C|H|,..

+ [(\ßA*A],V)i Jy

For the second weak equation let <p = 0 and a = *d\ — dsX for A = <f> txg

with (ß E T in the function space Cf(M, g) or ^(M^) corresponding to

the vector field X G C°°(M, TF). If the boundary conditions for a E EA

are satisfied, then we obtain with d = dy

i,p

/<JM

(ß, d^dMA)

/ f(ß, *d*dA-92A-*(9s*)9sA)Js1 Jy

/ / ( ß, *d^ct — *[A A *dA] + *dAds\Js1 Jy

+ Vsct - [$, cU] - Vs * dA - *(as*)ösA )

< C||A||jyi,«* + f /(C,d>)Js1 Jy

+

s1 Jy

ß, *dAds\ — Vs * dA )

< w^i*

Here we have used the identity

*dAds\ - Vs * dA = *[A A dsX] - [$, *dA] - (9s*)dA.

Moreover, we have used partial integration with vanishing boundary term

*ct|ay = 0 to obtain

/ f((,dAa)= [ f(dA(,*d\-ds\).Js1 Jy Js1 Jy

Now let X G C°°(M, TF) be perpendicular to the boundary dM = S1 x dY,then for all (ß G Q°(M) the boundary conditions for a = *dA — dsX E EA

107

are satisfied. Indeed, on the boundary dM = S1 x dY the 1-form A = (ß txg

vanishes, we have iXg = h r*dt for some smooth function h, and moreover

dcß = -§£-r*d£. Hence

*Ct|dY

a | ay

= dA|ay - *<9sA|ay = 0,

= *dA|ay - dsX\dY = -j£h * (r,dt A ndt) = 0.

Thus for all vector fields X G C°°(M, TF) that are perpendicular to the

boundary, lemma 3.3 asserts that ß(X) E W1,q(M,g). In particular, this

implies IT^-regularity of ß on all compact subsets K C int M. So it re¬

mains to prove the regularity on the neighbourhood r(S1 x [0, ^) x E) of the

boundary dM. In these coordinates we decompose

T*ß = £dt + ß.

Now firstly, lemma 3.3 applies as described above to assert the regularity

f = ß(r*dt) or E W1'q(fl x E,fl) on fl := S1 x [0, |A). Here a vector field

X that is perpendicular to the boundary is constructed by cutting off r*dt

outside of r(fl x E).So it remains to consider ß E Lq(fl x E, T*E <g> fl) and establish its W1,q-

regularity. In order to derive a weak inequality for ß on fl x E from (7.8) we

use à = T*(</xis + ißdt + a) with <p E Cs°(fl x E,g), iß E C$°(fl x E,fl), and

à E Wx*(fl x E, T*E ® fl) such that â(s, |A, •) = 0 and a(s, 0, •) G TAs£ for

all s G S1. This à satisfies the boundary conditions for (7.8) and it can be

extended trivially to a IT1,p-regular 1-form on all of M. Thus we obtain

/JnfixE

<

( Vsâ + *Via - dA(p - *dAtß , ß )

f (-Vs^ + V^-*dAct,e) + / (V^ + Vt^-d^â,C)JnxT JnxT

+ C||^ds + ^dt + o;||(?*.

Here we have introduced the decomposition r*A = $ds + \M£ + A, where

A E C°°(fl x E, T*E <g> fl) with A(s, 0) = As E £ for all s G S1. We have also

used the notation V^ = dt<p + [^f,<^], and moreover d^ and * denote the

differential and Hodge operator on E. Now firstly put à = 0, then we obtain

108 CHAPTER 7. FREDHOLM THEORY

for all (p,iß E Cs°(fl x E, fl) by partial integration

(<W, ß\nxT

*dAiß, ß ;nxE

< (C + ||Vt£- VsC||L9(nxE))||^llL9*(nxE)5

< (C + || Vs£ + VtCWmnxT))HV'll^(nxE)-

This shows that the weak derivatives dE/3 and dE/3 are of class Lq on fl x E.

So we have verified the assumptions of lemma 6.11 for ß and conclude that

VE/3 also is of class LP. So it remains to deduce the /^-regularity of dsß and

dtß on S1 x [0, j]xE from the above inequality for cp = iß = 0, namely from

( Vsct + *Vta, ß )fixE

< (C+||dAC + *d^l|L9(nxE))||â||L,.(nxE). (7.9)

This holds for all à E W1'p(fl x E,T*E <g> ß) such that â(s, |A, •) = 0 and

â(s, 0, •) G TAs£ for all s G S1. We now have to employ different arguments

according to whether q > 2 or q < 2.

Case q > 2 :

In this case the regularity of dsß and dtß will follow from theorem 5.4 on the

Banach space X = Z/9(E, T*E ® fl) with the complex structure given by the

Hodge operator on E with respect to the metric gSjt- From (7.9) one obtains

the following estimate for some constant C and all à as above:

//Jn Jt

ß, d8à + dt{*à)) <C\\â\ Li*(n,x*)- (7.10)

Note that this extends to the W1,q* (fl, L9*(E))-closure of the admissible à

from above. In particular the estimate above holds for all à E W1,q(fl,X)that are supported in fl and satisfy ct(s,0, •) G TAs£ for all s G S1. To

see that these can be approximated by smooth à with Lagrangian bound¬

ary conditions one uses the Banach submanifold coordinates for £ given bylemma 4.1 as before. Here the Lagrangian £ C X is the //^-restriction

or -completion of the original Lagrangian in .A°'P(E). It is modelled on

Wz,q(T,,g) x Rm as seen in chapter 4. However, in order to be able to applytheorem 5.4 (i), we need to extend this estimate to all à G C°°(fl,X*) with

supp et C fl and ct(s, 0) G (^T^yC)^ for all s G S1. This is possible since

109

any such à can be approximated in W1,q* (fl, X*) by àt G C°°(fl,X) that are

compactly supported in fl and satisfy the above stronger boundary condition

â,(s,0) G T^£ for all s G S1.

Indeed, lemma 5.8 provides such an approximating sequence a% without

the Lagrangian boundary conditions. From the proof via mollifiers one sees

that the approximating sequence can be chosen with compact support in fl.

Now for all s G S1 one has the topological splitting X = TAs£ © *TAs£and thus X* = (TAs£)L © (*Ty4s£)±. Since q > 2 the embedding X m-

X* iThens continuous. This identification uses the L2-inner product on X

which equals the metric u(•,*) given by the symplectic form u and the

complex structure *. So due to the Lagrangian condition this embedding

maps TAs£ ^ (*TAa£)-L and *TAs£ m- (TAa£)L- We write a = 7 + 5 and

ct« = 7« + ^ according to these splittings to obtain 7, 5 G C°°(Q,X*) and

7,, <J, G C°°(fi, X) such that *TA£ 3 7, ->• 7 G (TA£)X and TA£ 3 S, ^ S E

(*Ta£)x with convergence in W1,q*(fl,X*). The boundary condition on à

gives 7|i=o = 0. Moreover, dt'j is uniformly bounded in X*, so one can find

a constant C such that ||7(s, t)\\x* < Ct for all t G [0, f A] and hence for

sufficiently small e > 0

11 11 c i+—

\n\\Lo*(S^x[0,e],X*) < T^£q*

Now let S > 0 be given and choose 1 > e > 0 such that ||7||L«*(5ix[o,e],x*) < e(^

and ||7||vpi,«*(,six[o,e],x*) — ^- Next, choose i GN sufficiently large such that

Wli ~ l\\wl'i*{n,x*) < £^i and let /i G C°°([0, |A], [0,1]) be a cutoff function

with h(0) = 0, h\t>£ = 0, and \h'\ < f. Then â, := h^t + 6% E C°°(fi,X)satisfies the Lagrangian boundary condition ct?(s,0) G TAs£ and approxi¬mates â in view of the following estimate,

||â, - â\\wi,q* {n,x*)< ||/i(7, - l)\\w^*{n,x*) + 11(1 - ^)7llw^*(n,x*)< Il7* - iWw^i*(n,x*) + f ||7* - 7lL«*(n,x*)

+ ll7llw1.«*(S1x[0,e],X*) + ell^Hi«*(S'1x[0,e],X*)<65.

This approximation shows that (7.10) holds indeed true for all â E C°°(fl, X*)with supp et C fl and ct(s, 0) G (*Ta3£)~l for all s G 51. Thus theorem 5.4

asserts that ß E W^q(K,X) for K := S1 x [0, f ], and hence dsß and <9t/3 are

of class LP on S11 x [0, |] x E as claimed.

110 CHAPTER 7. FREDHOLM THEORY

Case q < 2 :

In this case we cover S1 by two intervals, S1 = L1 U L2 such that there are

isometric embeddings (0,1) °->- S1 identifying [j, |] with L1 and L2 respec¬

tively. Abbreviate K := [\, §] x [0, f ] and let fl' C (0,1) x [0, f A] be a

compact submanifold of the half space HI such that K C int fl'. Then for

each of the above identifications Sl \ {pt} = (0,1) one has /^-regularity of

ß on fl' x E by assumption and of *dAÇ + d^C from above. Now the task is

to establish in both cases the //^-regularity of dsß and dtß on K x E using

(7.9). For that purpose choose a cutoff function h G C°°(M., [0,1]) supportedin fl' such that h\x = 1. Then it suffices to find a constant C such that for

all 7 G C(fl' x E, T*E ® fl) (these are compactly supported in int(fi') x E)

/ <

Jn'xT\dsl,hß) <C|l7l

This gives //^-regularity of the weak derivative ds(hß) and hence of dsß on

K x Y,. For the regularity of dtß one has to replace <9S7 by dtf, then the

argument is the same as the following argument for dsß.We linearize the submanifold chart maps along (A;)se(o,i) £ £ H «4(E)

given by lemma 4.1 (iii) for the Lagrangian £ C „4°'9*(E). Note that this

uses the Lq*-completion of the actual Lagrangian in „4°'P(E). Abbreviate

Z := W^,q* (T,,g) x Rm and let *S;t denote the Hodge operator on E with

respect to the metric gsj- Then one obtains a smooth family of bounded

isomorphisms

eS)t: ZxZAL«*(E,rS®s) =: X

defined for all (s, t) G fl' by

®s,t(Ç>v'C>w) = dAsç + 2^1^1(0) + *s,dAsc + TZiwl *s,t,yi{s)-

Here 7, G C°°((0,1) x E,T*E <g> fl) with 7,(s) G TAs£ for all s G (0,1).Abbreviate Z°° := C2°°(E,fl) x Rm C Z, then QS;t maps Z°° x Z°° into the

set of smooth 1-forms Q^Ejfl). So given any 7 G ^(fl' x E,T*E <8> fl) we

have / := O"1 o 7 g C^(fl', Z°° x Z°°) and for some constant C

\\f\\Li*(n',zxz) < C|l7llL«*(n',x) = C|l7lL«*(n'xE)-

Write f = (fi, f2) with /, G C%°(fl', Z°°) and note that /n, dsf\ = 0 due to

the compact support. So one can solve A^/^i = dsf\ by (ßi E C^(fl', Z°°)

Ill

with /n, fa = 0 and An,cß2 = dsf2 by cß2 G Q°(^', ^°°). (For the Cf(E,g)-component of Z°° one has solutions of the Laplace equation on every fl' x {x}that depend smoothly on x G E.) Now let $ := ((ßu<ß2) E C°°(fl',Z x Z)and consider the 1-form

a~, h Q(-ds$ + Jodt$) E C°°(fl',X).

This extends to a 1-form onilxE that is admissible in (7.9). Indeed, ct7

vanishes for s close to 0 or 1 and thus trivially extends to s G S1. The

Lagrangian boundary condition is met since for all s G S1

â7(s, 0) = h(s, 0) • Os,o(-<9s^i - dt(ß2, -dscß2 + dt(ßi) G Qsfi{Z, 0) = TAs£-

So (7.9) provides a constant C such that for all â^ of the above form

/ (ß, <9sâ7 + <9t(*o:7))Jn'xT

< C\\â7\\Lq*çnx

Moreover, one has for all 7 G 0^(0,' x E,T*E <g> fl) and the associated /, <f>

and àry and denoting all constants by C

W^-iWli* {n,x)< Cll^llwn^Cfi'.ZxZ) < C\\f\\Lq*(n^ZxZ) < C||7||Lg*(n/xE).

Here the second inequality follows from A^'$ = dsf and lemma 5.7 (iii) and

(iv) as follows. In the Mm-component of Z, this is the usual elliptic estimate

for the Dirichlet or Neumann problem; for the components in the infinite

dimensional part F := IT.,1'** (E, fl) of Z (still denoted by <f>% and fß) this uses

the following estimate. For all iß G C^(fl', Y*) in the case i = 1 and for all

iß E C^°(fl', Y*) in the case i = 2

/ <Jn'xT

((ßi, &n,iß] (Aß/0,, iß)n'xT

( fi, dsiß )'n'xT

Now a calculation shows that

( dsft, iß )n'xT

< llAllL«*(n',Y)llV'l|w'1.9(n',y*)-

0sâ7 + ôt(*â7) = h-Q(A$)+ds(h-e)(-ds$ + Jodt$)+dt(h-Q){dt$-J0ds$)

112 CHAPTER 7. FREDHOLM THEORY

We then use A$ = dsf to obtain, denoting all constants by C,

{h-ß,dsl\n'xT

s <Jn'xT

(ß,h-Q(A$) + h-daQ(f))

< (ß, <9sct7 + 9f(*ô7)n'xE

+ C||/3||L,(n/,x.)(|| - ds$ + Jodt$)\\L«*(n',zxz) + \\f\\L«*(n',zxz))< C||7llL«*(n'xE)-

This proves the F?-regularity of dsß (and analogously of dtß) on S1 x [0, -f ] x Ein the case q < 2 and thus finishes the proof of the lemma.

Proof of theorem 7.1 :

Lemma 7.2 and the subsequent remark imply that for some constant C and

for all (a, <p) in the domain of -D(a,*)

||(ck,</?)||wi,p < C(||D(A)*)(a,¥?)||p + ||(a,¥>)||p).

Note that the embedding IT1,P(X) <—y LP(X) is compact, so this estimate

asserts that kerF(^$) is finite dimensional and imD^^j is closed (see e.g.

[Z, 3.12]). So it remains to consider the cokernel of D(a,*)- We abbreviate

Z := Lp(Sl x F,T*F ® fl) x LP(S1 x Y,g), then coker d{a^ = Z/imD{A^is a Banach space since imö^) is closed. So it has the same dimension as

its dual space (Z/imD{A,<s>))* ^ (imD^))1. Now let a : S1 x Y ->• S1 x Y

denote the reflection a (s, y) := (—s,y) on S1 = R/Z, then we claim that

there is an isomorphism

(irnö^))1(AC)

ker Da* {A,s>)

(ßoa,Coa).(7.11)

Here Da*(A,$) = D(A',&) is the linearized operator at the reflected connection

a*A = ^4'+$'ds with respect to the metric a*g on X. Note that ker Da*(A,®) is

finite dimensional since the estimate in lemma 7.2 also holds for the operator

Do-*(a,$)- So this would indeed prove that coker DA is of finite dimension and

hence DA is a Fredholm operator.

113

To establish the above isomorphism consider any (ß,C) E (imD^j)1,that is ß G LP* (S1 x F,T*F <g> g) and C e LP>* (S1 x Y,g) such that for all

a E E1/ and <p E IT1'P(51 x Y,g)

[ (D{A^(a,<p), (ß,O) = 0.

Js1xY

Iteration of lemma 7.3 implies that ß and Ç are in fact IT1,p-regular: We start

with q = p* < 2, then the lemma asserts Wl,p*-regularity. Next, the Sobolev

embedding theorem gives Lqi -regularity for some ci G (|,2) with ci > p*.

Indeed, the Sobolev embedding holds for any qt < -^rr, and | < ^-j- as well

as p* <j^-t

holds due to p* > 1. So the lemma together with the Sobolev

embeddings can be iterated to give Lq%+1 -regularity for ql+i = -^- as long as

4 > qt > 2 or 2 > qt > p*. This iteration yields q2 E (2, 4) and q% > 4. Thus

another iteration of the lemma gives W1,93- and thus also //-regularity of ßand Ç. Finally, since p > 2 the lemma applies again and asserts the claimed

IT1'p-regularity of ß and Ç. Now by partial integration

0= / {D{A^(a,ip),(ß,Q)Js^xY

= [ f (Vsa, - dAip + *dAa,, ß) + f [(V8<p-dAa,Ç)Js1 Jy Js1 Jy

(a,-Vsß-dAC + *dAß) + [ [(<p,-VsC-dAß)s1 Jy Js1 Jy

[{a Aß) - f [(<p,*ß). (7.12)s1 Jt Js1 Jt

Testing this with all a E ^(S1 x F, T*F <g> fl) C E1/ and <p> G C^(SX x Y, g)implies —Vs/3 — d^C + *dyi/3 = 0 and —VSC — d*Aß = 0. Then furthermore

we deduce *ß(s)\dY = 0 for all s G S1 from testing with tp that run throughall of C°°(5'1 x E, fl) on the boundary. Finally, fgl Js( a A ß) = 0 remains

from (7.12). Since both a and ß restricted to S1 x E are continuous pathsin „4°'P(E), this implies that for all s G S1 and every a G Ta3£

0 = / (aAß(s)) = uj(a,ß(s)),

where uj is the symplectic structure on „4°'P(E) from (4.1). Since Tas£ is

a Lagrangian subspace, this proves ß(s)\dY £ ^as£ for all s E S1 and thus

114 CHAPTER 7. FREDHOLM THEORY

ß E EA, or equivalently ß o a G EAoa- So (ß o a, ( o a) lies in the domain

of Do-*(A,*)- Now note that a*A = A o a — ($ o cr)ds, thus one obtains

(ß o a, £ o a) G ker Da*(A,$) since

^•(^)(j8off,C°ff) = ((-Va/5-dAC + *d^)o<T,(-VsC-d^)oa) = 0.

This proves that the map in (7.11) indeed maps into ker Da* (^ $). To see

the surjectivity of this map consider any (ß,() G ker Da*(A,<s>)- Then the

same partial integration as in (7.12) shows that (ß o a, Ç o a) E (im ^(yi^))-1-,and thus (ß,() is the image of this element under the map (7.11). So this

establishes the isomorphism (7.11) and thus shows that F^^) is Fredholm.

D

Chapter 8

Bubbling

In this chapter we discuss the possible bubbling phenomena for sequences of

anti-self-dual instantons with Lagrangian boundary conditions and bounded

Yang-Mills energy. So as in chapter 6 let (X, r, g) be a Riemannian 4-manifold

with a boundary space-time splitting and consider a G-bundle over X for a

compact Lie group G. Let (A1')^^ C A(X) be a sequence of solutions of

*FA + FA = 0,

<A|{s}xEj G A \/sESui = l,...,n.K'}

Suppose that this sequence has bounded Yang-Mills energy,

E(AV) := f \FA«\2 < F < oo V^N.Jx

This means that the curvature of the connections is globally L2-bounded.

If the curvature was in fact locally Lp-bounded for some p > 2, then the

compactness theorem B would imply that modulo gauge there exists a Cc-convergent subsequence of the connections. So in order to compactify the

moduli space of solutions of (8.1) with bounded energy, it remains to analysethe possible local blow-up of the curvature. We first discuss the standard

approach of local rescaling at a point x E X where the curvature blows up.

This is the situation of the following lemma, in which we use a local trivi-

alization and geodesic normal coordinates near x - either in R4 if x G int X

or in H4 = {y E R4 | yo > 0} if x G dX. In the latter case due to the

compatibility of the metric with the space-time splitting the coordinates can

be chosen such that the slices {y0 = 0, yi = c} correspond to the time-slices

of the boundary dX.

115

116 CHAPTER 8. BUBBLING

Lemma 8.1 Let x E X such that sur>,/lzN\\Fa» \\T^rn , ^

= oo for all e > 0.

Then there exists a subsequence (again denoted (A")), a sequence xP —y x,

and sequences 0 < ev —y 0, Rv —> oo such that svRv —y- oo and

\FAv{x")\ = (R")2, sup \FA,(y)\ < 4(/T)2.

(i) Lf x E int X or x E dX and lim sup^^.^ RvXq = oo, then there exists

a subsequence (again denoted (Av)) and possibly smaller 0 < ev < Xq

such that still rv := £VRV —> oo. Define a sequence of embeddings

4>u : Bru —y X in the coordinates near x by <ßv(y) := xv + -^y. Then

the rescaled connections Av := (ß*vAv E A(Brv) are anti-self-dual with

respect to the metrics gv := (RP)~2 <ß>*vg. Now for a further subsequencethere exist gauge transformations vP E G(Brv) such that

uu*Ä" —y A°° E A(R4)

converges uniformly with all derivatives on every compact set. The

limit A°° E A(R4) is a nontrivial anti-self-dual instanton on R4 of

finite energy £(A°°) < E, hence

lim £{Av\Bev{xV)) = lim £(Ä") = S (A00) > 8vr2.

(n) Lf x E dX and lim sup^,^ RPx^ = A < oo, then there exists a sub¬

sequence (again denoted (A")) such that the ev can be replaced by

\J(£V)2 - (xq)2 > 0 and still rv := evRP -> oo. Define the embed¬

dings (j)v : Dv := Brv n H4 —y X, metrics, and rescaled connections

Av E A(DV) as in (i) with the exception that [(ß„(y)]o := -j^yo- Fix

p > 4, then for a further subsequence there exists a sequence of gauge

transformations uv E G1,P(DV) such that

vv*Äv -± A°° E Aoc(H4)v—>oo

converges Wx'p-weakly on every compact subset. Moreover, the con¬

vergence is uniformly with all derivatives on every interior compact

subset ofH4. The limit A°° E ^(M4) n ^(intM4) is an anti-self-dualinstanton of finite energy £(A°°) < E and with weakly flat boundaryconditions on each time-slice {y0 = 0, y\ = c}. Ln case A > 0 it is

necessarily nontrivial. Moreover,

Kmmp£{Av\Bev{xV)) = limsup£(iv) > £(A°°).v—»oo v—>oo

117

Before we give the proof let us point out the significance of this lemma.

In case (i) the rescaled and gauge transformed connections converge to a

connection that extends to an anti-self-dual instanton on S4 = R4 fl {oo}.One then says that 'an instanton on S4 bubbles off at x\ Note that this

can happen in the interior as well as on the boundary of X. If case (ii) was

ruled out by some reason, then this would show that a minimum energy of

87T2 concentrates at every blow-up point of the curvature. Hence for a suit¬

able subsequence of the connections this bubbling off only happens at finitely

many points. On the complement of these points, the curvature is locallyF°°-bounded. Now theorem B applies on this complement (in the case of

bubbling at the boundary, one has to take away the full time-slice in which

the bubbling point lies). It asserts that a further subsequence of the connec¬

tions converges modulo gauge with all derivatives on every compact subset

of this complement of the bubbling points. At the interior bubbling points,Uhlenbeck's removable singularity theorem, proposition 8.3, below asserts

that the limit connection extends over the bubbling point - to a connection

on a different bundle. (The gauge transformation in the proposition can be

seen as transition function between the trivialization of the original bundle

on a punctured neighbourhood of the bubbling point and the trivialization

of the new bundle on the full neighbourhood.) So if there was only interior

bubbling, then this would provide the desired compactification of the moduli

spaces of solutions of (8.1) of bounded energy. However, the bubbling at the

boundary poses two difficulties.

Firstly, in the case of instantons on S4 bubbling off at the boundaryone needs a generalization of Uhlenbeck's removable singularity theorem to

points at the boundary.

Secondly, if case (ii) in lemma 8.1 occurs, then one does not even have a

minimum energy concentrating at the blow-up point of the curvature. This is

due to substantial flaws of the local rescaling: By cutting out only small balls

in each time-slice of the boundary one loses the global Lagrangian information

in the boundary condition. The slicewise flatness at the boundary does not

suffice to make the boundary value problem elliptic and hence obtain higher

regularity at the boundary. In the case A > 0 one still obtains a nontrivial

instanton in the limit due to the C°°-convergence in the interior of the half

space. For such instantons Salamon conjectured in [Sal, Conjecture 3.2] a

quantization of the energy that would imply that this instanton on the half

space bubbles off with a minimum energy of 87T2. In case A = 0, however, one

does not have C°-convergence of the curvature at the rescaled blow-up points,

118 CHAPTER 8. BUBBLING

so the limit connection might be trivial and thus have no energy althoughsome energy bubbles off 'very close to the boundary'.

So the results of the standard bubbling analysis in lemma 8.1 are far

from a complete understanding of the possible bubbling phenomena for the

boundary value problem (8.1). We still indicate its proof in the followingand give a further discussion of the bubbling at the boundary in a subsec¬

tion below. Lemma 8.1 uses Hofer's trick, [HZ, Ch.6,Lemma 5], as well as

Uhlenbeck's removable singularity theorem, [Ul, Theorem 4.1] (which holds

for Yang-Mills connections as well as for anti-self-dual connections).

Lemma 8.2 (Hofer's trick)Let f : X —y [0, oo) be a continuous function on a complete metric space X.

Assume that x0 E X and e0 > 0 are given. Then there exists x E B2eo(x0)and e E (0, e0] such that

£f(x) > e0f(x0) and sup f(y) < 2f(x).yeBe(x)

Proposition 8.3 (Uhlenbeck's removable singularity theorem)Let A E „4(F4\{0}) be an anti-self-dual connection over the punctured 4-ball.

Lf £(A) < oo then there exists a gauge transformation u E G(B4 \ {0}) such

that u*A extends to a smooth anti-s elf-dual connection over B4.

Proof of lemma 8.1 :

Fix sequences 0 < evü —y 0 and Rq —y oo such that £qRo —> oo. By assumption

one finds a subsequence (again denoted (^4i/)) and x^ E Bev(x) such that

\FA«(xo)\ > {Ro)2- Now apply Hofer's trick to the function y/\FAA on X

for each v E N to obtain the required xv E B2s^(xq), ev E (0, £q], and

R" = y/\FAv(x")\.Next, note that the metrics gv smoothly converge to the Euclidean metric

on R4 since we used normal geodesic coordinates centered at x = limx". For

the rescaled connections one obtains FA„(y) = (Rl')~2FAu(xu + j^y). Hence

for the norms with respect to the metrics gv one has

|F^(0)|=1, ||^||Lao(D,)<4.

(In the rescaling of case (ii) only the first identity has to be replaced by

\FAv(tPPh , 0, 0, 0)1 = 1.) The energy, however, is preserved by this rescaling,

£{Av) = £{Av\Bev{ßV)).

119

Now in case (i) the domains of ÄP can be chosen as balls that exhaust R4. So

the strong Uhlenbeck compactness theorem for noncompact manifolds with¬

out boundary (see e.g. [We, Theorem 11.3] or its generalization, theorem B)applies and proves the convergence claimed in (i). The nontriviality of the

limit connection is due to

|F4co(0)| = lim \(uv)-1FAuuv(0)\ = 1.v—>oo

Next, R4 is conformally equivalent to S4 \ {pt}, so A°° can be seen as anti-

self-dual connection on the punctured S4 with finite energy. Uhlenbeck's

removable singularity theorem, proposition 8.3, then asserts that ^4°° extends

to an anti-self-dual connection .4°° on a bundle F —y S4. Hence its energy is

a multiple of the second Chern number of the bundle F over S4,

£(A°°) = i {FAooA*FAoo) = - (FAooAFAoo) = 8n2c2(P).Js* /s4

Finally, this energy is at least 87r2 since the Chern number is integral and

the limit connection A°° is nontrivial, so £(A°°) > 0.

In case (ii) the weak IT^-convergence is due to the weak Uhlenbeck

compactness theorem for noncompact manifolds, see e.g. [We, Theorem 8.5].This weak convergence preserves the anti-self-duality equation as well as

the slicewise weak flatness at the boundary, and it gives the bound on the

energy of the limit connection. The interior C°°-convergence again follows

from the strong Uhlenbeck compactness theorem. In case A > 0 this interior

convergence implies the nontriviality of the instanton since

|F4oc(A,0,0,0)| = lim (F^f/T, 0,0,0) | = 1.v—>-oo

D

Bubbling at the boundary

For the discussion of the possible bubbling phenomena at the boundary we

restrict our attention to the possible blow-up of the curvature at one space-

time slice r({pt} x E) of the boundary. We use the decomposition induced

by the extended space-time splitting f : U x E —> X. Here one can choose

U C HI = {(s, t) E R2 11 > 0} as a neighbourhood of 0 and then examine the

120 CHAPTER 8. BUBBLING

possible blow-up of the curvature at {0} x E. So instead of the Av above we

consider a sequence of connections

<Tds + Wdt + Av E A(U x E)

with $",#" eC°°(Ux E,fl), and A" E C°°(U x E,T*E ® fl) for all v E N.

The boundary value problem (8.1) then becomes

b: + *sr = o,

dt$"-day' + [tyv,$''] = *FAv,

^|(s,o)xEe£ V(s,0)eW.

Here we have introduced the following notation for the components of the

curvature (dropping the superscript v):

Bs := dsA - dA$, Bt := dtA - dA^.

The Bianchi identity then takes the form

VSFA = dABs, VtFA = dABt,

VsBt - VtBs = dA(dt$ - dsy + [tf, $]).

Moreover, the bound on the energy becomes

sup / IF^P + IF^I2 < ±E < oo.jj/|2 | | t? |2 ^ 1

j/GN J

Note that |FS| = \Bt\ and \FA\ = \dt$-dsty + [$, $]|. Now a first observation

is that | Fa | cannot blow up faster that |FS|, in particular if Bvs is Lp-bounded

near {0} x E for p > 4, then there is no bubbling at all. This is made precise

by the following lemma.

Lemma 8.4 Let i"-)(0,z)eWxS be such that

\FA»(x")\ = (R'/)2^oo.

Then for all p > 4 and 0 < ev —> 0 with evRP —y oo there exists a constant

c > 0 such that

WrPfRf *vi

><RU)~-II s \\LP{Bev (xv))

— v '

121

Proof: Let x%, Fq, and Eq be as supposed. Assume in contradiction that for

some subsequence and 5V —y 0

\m\LP{BS^o)) < s^(R^.

Use Hofer's lemma 8.2 for the functions \/\FAv\ to obtain xv E B2ev(xq),

e E (0, ±eg], and Ru := y/\FA»{xv)\ such that rv := evR" -+ oo,

\FA.(xv)\ = (F1')2, IIF^IU-^^)) < 4(Ä")2,

and one still has

irai^flX*")) < W)^ < V(Rvf^.

Then one can pullback the connections and rescale the metrics as in lemma 8.1

to obtain $"ds + Wdt + Ä" E A(D») with Dv = BrJ or Dv = Br» n H4 if

limsupt^F1" < oo. Then HF^H^ is bounded and Bvs converges to zero in

the F^-norm on every compact set. (Here one writes xv = (sv,f,zv) and

uses local coordinates for zu E E, so HI4 = {(s,t,z) E R4 | t > 0}.) Now

Uhlenbeck's weak compactness theorem on noncompact manifolds (e.g. [We,Theorem 8.5]) gives a subsequence of gauge equivalent connections that con¬

verge IT1,p-weakly on all compact subsets. The limit $ds + ^dt + A is a

connection on R4 (if lim sup fi?" = oo) or on H4. The convergence is ac¬

tually in C°° on every compact subset of the interior by Uhlenbeck's strong

compactness theorem for anti-self-dual instantons (e.g. [We, Theorem 11.3]).Furthermore, the proof of theorem B also provides a IT^-bound on the FAu.

(The proof only breaks down at the higher estimates in theorem 6.8 for the

s- and t-derivatives of the A", which require the Lagrangian boundary condi¬

tion.) So for a further subsequence, the curvature converges locally in C°, and

hence the limit connection is nontrivial, |Fa(0)| = 0 or |Fa(0, A, 0, 0)| = 1 if

\m\tuRl/ = A < oo. The limit connection moreover satisfies

dsA - dA$ = 0,

< dtA - dA^ = 0,

k

dt$ - dsy + [*, $] = *FA.

Here one can choose a gauge in which ^ = 0, then A is t-independent and

hence \FA(0, t, 0, 0) = 1 for all i > 0 in contradiction to the finite energy.

122 CHAPTER 8. BUBBLING

Indeed, an a priori estimate for Yang-Mills connections with small energy

asserts that the curvature must decay as t —y oo, see [Ul, Theorem 3.5] .

Now let us assume that ||F"(s,£)||lp(e) blows up at (s,t) = 0 for some

p > 4. By Hofer's lemma 8.2 one finds (sv,tf) —y 0 and 0 < eu —y 0 such

that rv := evRv -> oo and

\\Bvs(s\f)\\LPm = R" -+ oo, sup ||^(s,t)||LP(s) < AR". (8.2){s,t)eB2£V{sv,tv)

i

Here we apply the Hofer trick to the functions ||-B^||!^fS), so we in fact even

have ev\[RP —y oo. Then as a direct consequence of the above lemma one

has a bound on the rate at which FAv can blow up.

Corollary 8.5 Suppose that (sv,tv) -)• 0, RP ->• oo, and 0 < ev -> 0 such

that ru := e"Rv -) oo and fS.£j. Then

(^-^IIF^H^ ^ )xS) y o.

Proof: Assume in contradiction that (for a subsequence) there exist 5 > 0

and z" G BeV(sv,tv) x E such that IF^^)) > (ôRv)2p-*. Since E is compact

one finds a further subsequence such that xv —> (0, z) for some z G E.~ P

Then lemma 8.4 applies with RP replaced by RP := (ôR")^-4. (Note that

evRP > £V\JHRP for sufficiently large v, and thus converges to oo.) Thus one

obtains for some constant c > 0 the following contradiction:

c5R" = c{RPf^ < HBril^BXx")) < 4R»Yol(B2£V(s»,r))lr.

D

Let us try a 2-dimensional rescaling only in the (s, t)-variables. Define

embeddings <f>v : Dv x E -> W x E for D" := [-r", r"] x [- min(rl/, tvRv), rv\by ^(cr, r, z) := (s" + ^7a, f + ^jr, z). (In the case lim sup VRP —y 00 this

will have to be adapted by replacing tv + -^r with jjp-r.) Then define the

rescaled metrics gv := g(sv + ^ju, f + ^jt, 2) on Dv x E and the followingconnections in A(DV x E):

$"ds + <Fdt + Av := ^* ($"ds + tf"d* + A")= (^$"ds + ^*"dt + >T) o fa. (8.3)

123

These now satisfy the boundary value problemx

dsÄv -dÄA" = -* {dtÄv - dAvW),dt®" - dß" + [*", $"] = (R")~2 * FÄ„,

Ä»\{s,_w)xvE£ MsE[-rv,rv].

Moreover, one has \\(dsÄv - dÄV$v)(0, 0)||lp(e) = 1 for all v E N and the

following estimates for some 5V —y 0

II^IU-^^^s)) < 5V(RV)^,\\dsÄ» - d^<r||Loc(D„,LP(s)) < 4, (8.4)

\\dt^ - ds^ + [^,^}\\Lo0{D,tLP{^) < S^R-)^-2 = 6v(RP)-f^.

Note that due to p > 4 the last norm converges to 0 as v —y oo. This hints

at a holomorphic curve in the space of connections as limit object. If the

connections would converge in some gauge, then in the limit one could find

a gauge such that $ and ^ vanish simultaneously, and thus be left with

dsA + *dtA = 0. This should yield a nontrivial finite energy holomorphiccurve in the complex Hilbert space (*4°'2(E), *). If lim sup^^ fRP = oo,

this plane should extend to a holomorphic sphere by a removable singu¬

larity theorem. Otherwise one would obtain holomorphic halfplanes with

Lagrangian boundary conditions that should extend to a holomorphic disc

with Lagrangian boundary conditions. There are two indications that this

might be the right approach to the bubbling phenomena at the boundary.

Firstly, one expects for general reasons that there should not be any

holomorphic spheres bubbling off. All spheres in the space of connections

are contractible, so any holomorphic sphere u : S2 —y *4°'2(E) (extendingto ü : B2 —y ^t°'2(E)) has zero energy Jg2 \dsu\2 = Js2 u*u) = Jß2 ü*du = 0.

Moreover, if the bubbling off of holomorphic spheres was possible, then this

should also happen at interior E-slices, where so far all blowing up of the

curvature has been explained as bubbling off of instantons on S4. In fact,the 2-dimensional rescaling does not lead to holomorphic spheres (tvRv is

always bounded) if one excludes the cases where instantons on S4 bubble off

at the same boundary slice (or similarly at the same interior slice), as will

1This can be seen as anti-self-duality equation, where the metric on S is conformallyrescaled by the factor (Rv)2. An analogous equation without boundary conditions was

studied in [DS2, (3.5)], where the roles of A and $ds + $dt are interchanged.

124 CHAPTER 8. BUBBLING

be shown in the subsequent lemma 8.6. Secondly, the convergence of the

rescaled connections to a holomorphic curve in the space of connections can

be made rigorous if one assumes an L°°-bound on FAv near the boundary,see lemma 8.8 below.

So firstly, assume that there is no instanton on S4 bubbling off at {0} x E

(as described in lemma 8.1 (i)). This is equivalent to the assumption that for

all subsequences of the connections (still denoted (A")) and all (s^^") —y 0

there is a bound

sup (02|| (FA* + B: A ds) {8", OL^b.c^xe) < oo.

Under this assumption the following lemma proves that in the above 2-

dimensional rescaling one always finds limsupti/Fi/ < oo, which would giverise to a holomorphic disc with Lagrangian boundary conditions - not a holo¬

morphic sphere. (In fact, this lemma also holds if (su,t") converges to some

interior point (s, t) with t > 0, and that shows that there are no holomorphic

spheres bubbling off in the interior.)

Lemma 8.6 Let (sv,tu) —> 0 and suppose that

SUp (t17) \\FA<'\\l°°{BUv{sv,f)xT) < OO.

J/GN 2*

Then there exists a constant C such that for all v E N

(02||F^r)|U(s)<(7.

Proof: In a sufficiently small neighbourhood of (0, 0) one can assume that the

metric on E is independent of (s,t). (One only gets small additional terms

below that can be absorbed into others.) Then use the Bianchi identities

and the anti-self-duality equation to calculate (dropping the superscript v)V'SBS + VjFj = — *dA* FA and thus

(V2 + V2t)Bs = Vs(-VtF* - *dA * FA) + Vt(VsFt - d^ * FA)= (ViVs - VsVt)Ft - *(VsdA - d^V,) * FA - *dA * WSFA

- (Vtd^ - dAVt) * FA - dA * VtFA

= [*FA, Bt] - *[BS, *FA] - *dA * dABs - [Bt, *FA] -dA* dABt

= -AAB8-3*[Bs,*FA].

125

Fix any z E E and abbreviate vP := |F^|2. This satisfies on B\tv(sv,f ,z)with some constant C

Au" = |Wf + 2( Bs, (V2 + V2 + AA)BS )= |Wf+ 6([FS A FS],FA)> -C(?)-2uu.

Hence the a priori estimate in lemma 8.7 implies

uv(sv,tv,z) < p (|C2 + 16)(t")-4 f uv < C'(t»)-4E.Jbl (s",r,z)

2f

This proves the lemma since the constant C is independent of z E E.

Here we have used the following a priori estimate which is an adaptationof [DS2, Lemma 7.3] to 4 dimensions.

Lemma 8.7 Let u : R4 D Br —y [0, oo) be a C2-function such that for some

constant c > 0

Au > —cu.

Then

fi«(0)< (|c2 + r"4) j u.

Proof: The general case r > 0 of this lemma can be reduced to r = 1 by

rescaling, so it suffices to prove the lemma for r = 1. Consider the followingfunction / : [0,1] -> R,

f(p) = (1 ~ p)4supu.Bp

Choose p* E [0,1) at which / attains its maximum, let S := |(1 — p*), and

choose w* E Bp* such that supB tu = u(w*) =: c*. Then for all w E Bp*+Ô

one has u(w) < 16c* since

±(l-p*)4u(w) < (l-(p*+5))4 sup u = f(/+5) < f(p*) = (l-p*)V.Bf,*+8

Now ü(w) := u(w) + 2cc*\w — w*\2 is a subharmonic function on B$(w*),hence for all 0 < p < 5

c* = ü(w*) < -^7— I ü = \cc*p2 + -^7— I u.

n r Jbp(w*) k P Jbp(w*)

126 CHAPTER 8. BUBBLING

In case |cc*£2 > \ choose p2 = §c x< 52 to obtain

2 f 128 c2 f1*0 < C* < -^r— / U < /1 ^ " "

^PAJB(l{w*)-

9 TT2 JBl

In case |cc*£2 < | choose /? = 5 to find that

3?

u(0) = /(0) < /(,•) = 16 5V <-2

Putting this together yields the claim for r = 1,

g«(0) < (|c2 + l) [ u.

J Bi

D

Now assume that no instantons on S4 bubble off at the boundary slice

{0} x E, then we have shown above that only holomorphic discs can re¬

sult from the 2-dimensional rescaling. Under the additional assumption of

a bound on FAv near the boundary we now show that in fact one obtains

a holomorphic halfplane with Lagrangian boudary conditions. Its finite en¬

ergy should allow for a removal of the singularity at infinity, giving rise to

a holomorphic disc. However, it is not yet clear whether this limit objectis necessarily nontrivial - this would require a stronger convergence in the

following lemma. 2

Lemma 8.8 Suppose that p > 4, (s", f) —y 0, and 0 < ev —y 0 such that

II^(^>OIIlp(e) = Rv -»• oo, sup \\Bvs{s,t)\\i*V) < ARV

(s,t)eB2£v(sv,tv)

with rv := evRP —y oo and fRv —> A < oo. Moreover, assume that for all

T > 0 there exists a constant Ct such that

sup ||F4,(s,t)||LP(s) < CT WeN.

\S S I, t 2^j^ït

Let Dv := [—rv, rv\ x [0, rv\ and define the rescaled metrics gv and connections

&ds + &dt + Ä1' on Dv x E as in (8.3) with F + ^t replaced by -^t. Then

2More precisely, one needs the convergence of Bvs in C°(H, LP(E)). This might follow

from an analogon of the curvature estimates in [DS2, Theorem 7.1].

/ u.

JbJw*)

127

one finds a subsequence and a gauge such that Qv —y 0 and ^ —y 0 converge

W1,p-weakly on every compact subset of HI := {(s,t) E R2 | t > 0}, and

i"->ie IT1'P(HI, „4°'P(E)). The limit Ä is a holomorphic curve in „4°'2(E)with Lagrangian boundary conditions and finite energy,

dsÄ + *dtÄ = 0, /",,„ 7l|2 ^ ^<~ F.[ \\dsÄ\\)

Jm^|(.,o)xe£ VsG '-""

,lL2(S)

Proof: Except for a shift in the t-variable, this rescaling is the same as in

(8.3). One then has \\{dsÄv - dÀ„$v)(Q,t"Rv)\\LPp) = 1 for all v E N, and

the additional assumption on Fa» yields the following estimates instead of

(8.4): For all F > 0

\\Fäv\\Lc°([-t,t]x[o,t],lp(y;)) < Ct,

\\dsAl/ - d^$i/||Loo([_r;T]x[o,r],Lp(s)) < 4,

||9t^_9s^ + [^,$-]||LOO <(7r(^)-2 _^ o.ii—>-oo

Now Uhlenbeck's weak compactness theorem on noncompact manifolds (e.g.[We, Theorem 8.5]) applies and yields a subsequence and gauges such that

those connections converge in the weak IT^-topology and the strong L°°-

topology on every compact subset of H x E. The limit «I? ds + ^> dt + A E

Aoc(HI x E) satisfies the following boundary value problem:

d8Ä-dÄ$ = -*(dtÄ-dÄÜ),dt$ - ds^ + [§, $] = 0,

%,o)xs e £ VseR.

One finds a further gauge transformation in IT^'^HI, IT1,P(E, G)) that makes

$ and ^ vanish and preserves the convergence of $" ,^fp'. This gauge more¬

over transforms A into a connection A E ITlof (H, ^t°'p(E)) that is holomor¬

phic with Lagrangian boundary conditions as claimed. However, the con¬

vergence of Av only is locally in the weak W1,P(M.,LP(T,))- and the strong

F°°(HI, Fp(E))-topology. Moreover, this rescaling and gauging preserves the

energy bound

f ||Mli,(E) < limsup f \\d.Ä" - d**"||£8(E)Jm i/^-oo Jh

< limsup/ IcVr-d^T < E.i/^-oo J B2ev{sv,t")xT,

D

128 CHAPTER 8. BUBBLING

Appendix A

Gauge theory

In order to set up notation and state some general facts that are used in this

thesis we give a short introduction to connections and curvature on principalbundles.

We consider a principal G-bundle tt : F —»• M, that is a manifold F

with a free right action F x G —y F, (p, g)-y pg of a Lie group G such that

the orbits of this action are the fibres Px = 7r~1(x) = G of a locally trivial

fibre bundle tï : F —> M. So M is a smooth manifold, the G-action preserves

the fibres, n(pg) = tt(p), and there exists a bundle atlas M = [JaeA Ua with

equivariant local trivializations

öTT"1 (£4) — uaxG

°"

P -" (AP):UP))'

More precisely, the <3?a are diffeomorphisms and their second component is

equivariant, <ßa(pg) = <j>a{p)g.This atlas gives rise to transition functions <ßaß : Ua fl Uß —y G defined

by ^o^-1^,^) = (x,(ßaß(x)g) for x EUanUß, i.e. <f>aß(x) = (ßa(p)<ßß(p)~1for all p E ti~1(x).

Now for any other manifold F with a representation a : G —y Diff(F)the associated bundle PxT is the set of equivalence classes [p, f\ in

P x F, where the equivalence is given by a, i.e. [p,f\ ~ \_pg,°(g~1)f\for all g E G. (Here we write |_-, -J for the equivalence classes in order to

distinguish this notation from the Lie bracket [•, •].) With the projection

ïï\_P,g\ = 7r(p) this is a principal bundle over M with fibre F. A local

trivialization $a : 7r_1(Fa) —y Ua x G of F naturally induces the following

129

130 APPENDIX A. GAUGE THEORY

local trivialization of F xa F:

= n-^Ua) —^ UaxFa'

[P,f\ — (vr(p),a(^(p))/)•

We will, for example, encounter the associated bundle where F is the group

G itself and the representation is the conjugation c : G —y Aut(G) given by

cg(h) = ghg-1 Vg,hEG.

Another example is flp :— P XAd fl- Here fl — T^G is the Lie algebra of G

and Ad : G —> End g, g \-y Ad9 — djcg is the adjoint representation of G on

its Lie algebra fl,

Adff(0 := «7&-1 VÇGfl,<?GG.

This uses the following notation: For £ G fl and g G G

gi := djLgd) = A|i=o£exp(t£) G T,G.

Here Lg denotes left multiplication by g and exp : fl —y G is the usual

exponential map (with respect to any metric on G). The notation £g is

defined analogously by right multiplication. This makes particular sense

when G C Cnxn is a matrix group since then gÇ can be understood as matrix

multiplication.

Moreover, the adjoint representation of fl on g is given by the Lie bracket

of vector fields as follows. We identify the Lie algebra elements £ G fl with

left invariant vector fields g \-y gÇ on G, then for £, £ G g

ad^(C) := diAd(0C = H^exp^OCexp^)-1 = ACW = [6C]-

In the case of a matrix group note that the Lie bracket is given by the

commutator [£, (] — ££ — ££.

Coming back to the principal bundle let 6 : G —» Diff(F), g \-y Qg denote

the action of G on P. Then for v E TpF, g E G we write

vg := dpeg(v).

Moreover, the infinitesimal action is defined for £ G fl and p G F by

pi := dj<d(i)p = 5j|t=0pexp(if)-

131

Furthermore, a G-bundle isomorphism is a bundle isomorphism that pre¬

serves the action of the Lie group G, and such isomorphic bundles are usuallyidentified. So when studying a fixed bundle we also have to consider its G-

bundle automorphisms, i.e. diffeomorphisms iß : F —y P such that tt o iß — ißand that are equivariant, iß(pg) — iß(p)g for all p E P, g G G. Every such au¬

tomorphism is given by iß(p) — pu(p), where the smooth map u : F —> G is a

unique element of the gauge group G(P)- This means that u is equivariant,

u(pg) = g~lu(p)g Vp G F, g G G.

Obviously, composition iß2 o ißt of G-bundle isomorphisms corresponds to

group multiplication U\u2 of the corresponding gauge transformations. More¬

over, the gauge group is isomorphic to the group of sections of the associated

bundle FxcG. Let u G G(P), then the corresponding section ü : M —y FxcGis given by

ü(tt(p)) = [p,u(p)\ MpE P.

In the local trivialization a gauge transformation u G G(P) is represented by

ua — 4>a ° ü ' M —y G and acts by (x,g) t->- (x,gua(x)) on Ua x G. Here

</>«( li^pj) = <f>a(p) #^a(p)_1 is the second component of the trivialization <3>a

of F xc G. Thus ua(x) = (ßa(p)'u(p)4>a(p)~1 f°r all P G 'K~1(x), and hence on

Ua n Uß one has the transition identity

uß = (f>äßUa(t>aß-

Finally, to introduce connections we first note that the G-bundle F has a

canonical vertical subbundle V C TF given as follows. For every p E P

the vertical space Vp — ker(dp7r) C TpF is composed of all tangencies p£,

£ G fl to the orbits of G through p. Every complement of Vp is isomorphicto im(dp7r) = Tn^M, but there is no canonical choice of this horizontal

space in TpF. Now a connection of F defines such an equivariant horizontal

distribution H C TF as follows.

A connection on F is an equivariant fl-valued 1-form with fixed values

in the vertical direction, i.e. A G Q4(F;fl) satisfies

Apg(vg) = g~lAp(v)g V«G TpF, g EG,

AM) = e Vp g F, i g g.

132 APPENDIX A. GAUGE THEORY

We denote the set of smooth connections by A(P). Now every connection

A G A(P) corresponds to a splitting TF = V © H, where the horizontal

distribution H is defined by Hp = kev Ap.Again, this can be formulated equivalently in terms of an associated bun¬

dle. If we fix one connection A G A(P) then the space of connections is the

affine space A(P) — A + £7Ad(F;fl). Here QAd(F;fl) denotes the space of

equivariant horizontal &-forms, i.e. r G Qk(P;g) that satisfy

Q*gr = g-1rg \f g G G,

hîTP = ° Vp G F, £ G fl.

Now this space is isomorphic to the space Qk(M;gp) of &-forms on M with

values in the associated bundle gP — P xAd0- Indeed, for r G fiAd(F;0) the

corresponding f G Qk(M;gp) is uniquely defined by

Lp, tp(Yy, ... Yk)\ = f7T{p)(dpn(Y1),... dpn(Yk)) VTi,... Yk G TpF.

Consequently, in the local trivialization every r G Q\d(P;g) is represented

by ra = (ßa of G nk(Ua;g). Here <^a(b^J) = <ßa(p) Ï (ßa(p)~l is the second

component of the associated trivialization of gP. On the intersection Ua fl Upof two charts these fc-forms satisfy

Tß = (ß~ßTa(ßaß- (A.l)

In the case of connections this local representation depends on the chosen

connection A and there is no canonical choice for this reference connection.

However, locally on 7v~l(Ua) a natural choice of the reference connection

is Aa — (ß^dißa .This corresponds to the pullback of the splitting under

$„ : P 4 [/„ x G. The local representative Aa E fi1(Fa;fl) of A E A(P) is

then given by

Aa(dpn(Y)) - (ßa(p) A(Y) (j>a(p)-1 - dpcßa(Y) (j>a(p)-1 VF G TPF.

Note that the transition between different coordinate charts is different from

(A.l) since the reference connection is not globally defined. On Ua D Uß one

has

Aß = <\>~aßAa<ßaß + C/3d<?W

One can think of this as the effect of a local gauge transformation. So next,

we discuss the action of the gauge group. Consider a gauge transformation

133

u G G(P) and the corresponding G-bundle automorphism iß : p \-y pu(p).Their action on a connection A G -4(F) is given by

u*A := iß*A = M"1^M + M"1dM.

This is the connection on iß*P = F that corresponds to the connection A on

P. Hence u*A and ^4 are viewed as equivalent connections - they are gauge

equivalent. Finally, the local formula for the gauge action is

(u*A)a = u~lAaua + U^dUa.

This shows that locally a gauge transformation can also be thought of as a

change of the trivialization.

Connections also induce covariant derivatives on associated vector bun¬

dles. In particular, a connection A E A(P) defines the following covariant

derivative on gP :

VAr(flp) — r(T*M<g>0P)

s i—y ds + [A,s].

Here and throughout, T(-) denotes the set of smooth sections of a bundle.

For X ETXM with Y E TpP such that dpTv(Y) = X this evaluates as

VAs(X) = [p,dps(Y) + [A(Y),s(p)}\ E(gP)x,

where on the right hand side s G T(flp) is understood as map from F to g.

By a standard construction this covariant derivative can then be extended

to VA : r((g>feT*M <8> gP) -+ T(®k+1T*M <8> gP) for all k G N as follows. Let

V be the Levi-Civita connection on M, then for a G r(<g>fcT*M <g> gP) and

X0,...,XkET(TM)

VAa(X0, ...,Xk) = VA (a(Xu .. .,Xk)) (X0) - a{VXoX1,X2, ...,Xk)

-...-a(X1,...,Xk_1,VXoXk). (A.2)

But the covariant derivative on gP can also be understood as the special case

k = 0 of the exterior derivative

dA.rth(P;B) — OfôiPlB)

t i—y dr + [AAr].

134 APPENDIX A. GAUGE THEORY

Here [A A r] denotes the wedge product of the two forms with the Lie bracket

used to combine the values in g. For example, for A, B E Q\d(P;g) and

X, Y E TpP

[AAB](X,Y) = [A(X),B(Y)\ - [A(Y),B(X)\.

Now dA2 does not vanish in general, but we obtain dAdAT = [Fa A r] for all

r G fiAd(F;fl), with the curvature

FA = dA + ±[AAA] En2Ad(P;g).

The curvature satisfies the Bianchi identity (see e.g. [KN, II,Theorem 5.4])

dAFA = 0.

Locally, the exterior derivative dA on r G QAd(F; fl) is represented by

(dAT)a = dra + [AaAra].

Thus for the curvature in terms of the local representatives Aa of the con¬

nection we obtain the same formula as globally,

(FA)a = dAa + ±[AaAAa\.

In coordinates (x1,... ,xk) of Ua and dropping the subscript a one has thea d

dx* ' 8x3following formula for the components FtJ :— FA(-J^, öfy) of the curvature

F — Mi_

Ml _l [4 41

A change of the trivialization has the effect that on Ua fl Uß the local repre¬

sentatives of the curvature satisfy (FA)ß = (ßäß(FA)a(ßaß- Analogously, gauge

transformations act on the curvature by the adjoint action,

Fu*a = u~1Fau VAeA(P),ueG(P).

A connection A E A(P) is called flat if its curvature vanishes, FA — 0. This

is equivalent to dA ° dA = 0, and moreover one has the following characteri¬

zation, see e.g. [DK, Section 2.2].

Theorem A.l A connection A E A(P) is flat if and only if the associated

horizontal distribution H — kerA c TF is locally mtegrable. That is given

a fixed trivialization over a simply connected domain Ua C M there exists a

local gauge transformation u G G(P\ua) suc-h that (u*A)a — 0, or equivalently

if one changes the trivialization by ua, then H = TUa x {0}.

135

Every connection A G A(P) also defines parallel transport in F. Along a

path 7 : [0,1] —y M the parallel transport

n7 : F7(0) —y P7(i)

is given by s(0) i->- s(l), where s : [0,1] —y F is a horizontal lift of 7, i.e.

7T o s — 7, such that s(t) E H3^ for all t G [0,1]. The parallel transport is

equivariant in the sense that Yi^pg) = Ii1(p)g, so n7 is determined by its

value on one fixed p G P7(o)- F°r loops 7 starting at a fixed x and with fixed

p E Px, the parallel transport can be identified with group elements /i7 via

n7(p) — phj. However, if p is allowed to vary, then h7 is welldefined only up

to conjugation.If the connection is flat, i.e. H is locally integrable, then the parallel trans¬

port is invariant under homotopies of the path with fixed endpoints. Thus

for fixed x E M and p E Px every flat connection determines a representationof 7Ti(M, x) in G, the holonomy based at x,

7n(M,a;) — GPx'p[A)-

7 >— v

Variation of x in a connected component or another choice of p results in a

conjugation of pXtP. In the case of a trivial bundle F — M x G, there is a

natural choice p = (x, 1) which leads to a natural welldefined based holonomy

A(P) —^ Hom(7n(M,a;),G)PxZ

A — Px(A).

If M is connected then the conjugacy class of p := px>p is welldefined inde¬

pendently of the choice of x and p. Moreover, gauge transformations also act

on the holonomy by conjugation,

Px,P(u*A) = u(p)-1px,p(A)u(p).

One even finds that two flat connections are gauge equivalent if and onlyif their holonomies are conjugate. Conversely, given a representation of the

fundamental group in G one can construct a G-bundle and a connection on

it that has this holonomy map. So denote the space of flat connections on

a G-bundle over a fixed base manifold M by „4flat(M), the group of gauge

transformations on such bundles by Q(M), and the conjugacy equivalencerelation by ~, then this leads to the following observation (see e.g. [DK,Proposition 2.2.3]).

136 APPENDIX A. GAUGE THEORY

Theorem A.2 Let M be connected, then the holonomy induces a natural

bijection of sets

Aat(M)/0(M) = Hom(7T1(M),G)/ ~ .

Next, one would like to have a gauge invariant quantity measuring the

nonflatness of a connection. Note that the norm of the curvature is indeed

gauge invariant if fl is equipped with an inner product that is invariant under

the adjoint action of G. So from now on we restrict ourselves to compact Lie

groups G because of the following proposition. Its proof can be found in [K,Proposition 4.24].

Proposition A.3 Let G be a compact Lie group and let g be its Lie algebra.Then there exists an inner product (-, •) on g that is invariant under the

adjoint action of the Lie group,

<gtg-\g(g~l) = (£,() V£,C e g,g e g. (a.3)

Remark A.4

(i) The G-invariant inner product on fl moreover satisfies for all £, (, n E g

(ii) The G-invariant inner product on g induces a metric on G by

{X,Y)G:=(g-lX,g-lY) MX, Y E T,G.

In this metric the left and right multiplications are isometries of G.

Denote by expff the exponential map with base point g G G and set

exp := expj, then for all £ G g and g G G

exp9(#£) = #exp(£), exp(#-1£p) = g~l exp(£)#.

Moreover, the geodesies are the flow lines of the left invariant vector

fields, hence they are 1-parameter subgroups: For all s, t G R and £ G 0

exp((s + t)Cß) — exp(s£) exp(t£).

137

Here (i) follows from differentiating (A.3) with g — exp(try). In (ii) one has

used that for all £ G 0 and g G G both t h-)> expff(tg£) and 1i->- #exp(t£) are

geodesies with identical initial values. The same holds for t i->- exp(tg_1£g)and t \-y g~x exp(t£)#.

A flow line 7(t) satisfies 7(t) — 7(t)£ for some £ G 0. One checks the

geodesic equation V77 = 0 with Z(t) = ^(t)rj for all r\ E g,

#(V77, Z) = £j g(j, 7??) - \£z g(j, 7) - 0(7, [7,7*7])= £7(£, 77)-i£z<£, £}-<£, [£,??]) = 0.

Throughout this thesis every compact Lie group G is equipped with the

metric from proposition A.3 and remark A.4. Furthermore, fix a metric

on M. This defines a volume element dvolM and the Hodge operator * on

differential forms. Together with the inner product of g this moreover defines

an inner product on the fibres of ®kT*M ® gP for all k E N0 as follows. In

the first component of the fibre, ®feT*M, use the standard inner product on

T*M in each factor. On (gp)x — {|_P,£J | p G n~l(x),^ E g\ the G-invariant

inner product of g induces the welldefined

<IMJ>IMJ>*>:=<£>C>.

For a, t E Vtk(M;gp) this pointwise inner product equals the inner productof the local representatives aa,ra in every trivialization over Ua C M,

(a, r)AkT*M(S)Sp = *((7A*t)Sp = *(<JaA*Ta)g = (aa, ra )a*t*m®b-

In the second and third expression the values of the differential forms are

paired by the inner product idicated by the subscript. For example, for

0-valued 1-forms a — aidx1 + o~2dx2 and r = Tidrc1 + r2dx2 on R2 one has

(<tAt)s = ({<7i,T2)B- (<J2,T1)g)dx1 Adx2.

Usually the inner product is clear from the context, so we drop all subscripts.

Furthermore, * denotes the obvious Hodge operator on gP- or g-valued dif¬

ferential forms. (When written in local coordinates as a sum of productsof sections (or g-valued functions) and differential forms dxn A ... dxlk the

Hodge operator only operates on the differential form.)Now the curvature Fa can be viewed as a section of ®2T*M ®gP, so the

norm induced by above inner product defines a function |Fa| : M —y R that

can be integrated to give the Yang-Mills energy

yM(A) = I \FA\2 dvolM.JM

138 APPENDIX A. GAUGE THEORY

Due to the invariance of the metric (A.3) this functional on A(P) is gauge

invariant,

yM(u*A)=yM(A) VuEG(P).

Its extrema solve the weak Yang-Mills equation

/<Jm

(FA,dAß) = 0 VßEQ^M-^p). (AA)

Here O,1^) denotes the smooth 1-forms. When the base manifold M is com¬

pact and has no boundary then for smooth connections (A.4) is equivalentto the usual Yang-Mills equation d^F4 = 0. If the base manifold is allowed

to have boundary then (A.4) for smooth connections is equivalent to the

following boundary value problem:

d^ = °'(A.5)

*FA\dM — 0.

Here the operator d*A : Çlk(M;gP) —y Q/s_1(M;ßp) is the formally adjointdifferential operator of dA : f2fe-1(M; flP) —> Qk(M;gP) defined in the usual

sense: For uj E Qk(M;gP) and all ß E Qk~1(M;gP) compactly supported in

the interior of M

[ (dAu,ß)= [ (u,dAß).JM JM

From this one sees that d^ —

— (—l)^-^-1) * dA* on Q,k(M;gP), where

n— dimM, and locally for all u E Qk(M; gP)

(d»a = d*ua - (-l^-W*-1) * [Aa A *ua]. (A.6)

The weak and strong Yang-Mills equation are preserved under gauge transfor¬

mations. For the weak equation this is obvious from the gauge invariance of

the Yang-Mills functional - the extrema come in gauge orbits. For the strong

(i.e. pointwise) equation (A.5) one can check that d*u*AFu*A = u~1(d*AFA)u.So far this is all well defined since we only considered smooth connections.

The Yang-Mills energy might however be infinite if the base manifold M

is not compact. In that case there also are no natural Sobolev spaces of

connections. So for the rest of this appendix we assume both M and G to

be compact and explain how above concepts generalize to the appropriateSobolev spaces.

139

Firstly, for 1 < p < oo and k E N the Sobolev space of connections

Ah'p(P) :=Ä + Wk'p(M, T*M <8> gP),

is independent of the choice of a smooth reference connection A E -4(F).Only the corresponding Sobolev norm ||-||vi/'s>p on Wh'p(M, T*M®gP) depends

on A unless k — 0. This norm is defined as usual - using the Levi-Civita

connection on M and the above norm on ®lT*M ® gP. The Sobolev space

then is the completion of the space of smooth connections with respect to

this norm. In a local trivialization over U C M the connections in _41,P(F)are represented by 1-forms in

Ahp(U) :=Wl*(U,T*U®g),

and the corresponding Sobolev norm is the usual IT^-norm on this space.

The gauge action can also be defined for a suitable Sobolev space of gauge

transformations for kp > n. (The definition of this Sobolev space requiresthe choice of an atlas for G, but for kp > n it is independent of this choice.)

Gk'p(P) :=Wk'p(M,PxcG).

This space consist of all those gauge transformations u = s • exp(£), where

s G G(P) is smooth and £ G Wk,p(M, gP) is understood as equivariant map

£ : F —> g. In a trivialization over U C M gauge transformations in Gh,p(P\u)are represented by maps u E Gk,p(U), i.e. u = s exp(£) : U —y G with

s G C°°(U, G) and £ G Wk'p(U, g). For more details on these Sobolev spaces

see e.g. [We, Appendix B].These sets are Banach manifolds in the topological space of continuous

gauge transformations and they are actual groups with continuous group

operations. A proof of the following lemma can for example be found in [We,Appendix A].

Lemma A.5 Let k E N and 1 < p < oo be such that kp > n. Then group

multiplication and inversion are continuous maps on Gk,p(P), and the gauge

action is a continuous map

Gk'p(P) x Ak~1'p(P) —y Ak~1'p(P)(u,A) i—y u*A.

If the base manifold M is noncompact, then one considers the Sobolev

spaces -4lo'c(F) and G\oc(P) of locally IT^-regular connections and gauge

transformations respectively - these are required to be of class Wk,p on every

compact subset of M.

140 APPENDIX A. GAUGE THEORY

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Acknowledgements

Being stuck with estimates and sometimes even progress itself is a toughbusiness which I would not have survived without a good deal of help. So I

would like to thank

Dietmar Salamon for suggesting this great project, for his everlasting encour¬

agement, for lots of great discussions, and for teaching me everything from

adiabatic limits to preparing pesto sauce,

Dusa McDuff and Edi Zehnder for co-examining my thesis at such short no¬

tice and for still giving a lot of helpful suggestions,

the 'symplectic gang' in and around the ETH Zürich for a great working

atmosphere, for stimulating discussions, and for some glorious symplecticaction: Meike Akveld, Kai Cieliebak, Urs Frauenfelder, Rita Gaio, Ralf

Gautschi, Thomas Mautsch, Felix Schlenk, Fabian Ziltener, Joa Weber, and

again the boss, Dietmar Salamon,

my family and friends all around the world for their love, for many great

times, and for helping me through the hard times,

all people that have not found themselves in this list yet for their support

and sympathy - and for reading at least this page of my thesis,

the Swiss National Science Foundation for a research grant that gave me the

peace and time to finish this thesis,

the swiss mountains and lakes for blisters, sunburns, utter exhaustion, and

thus taking my mind off the estimates.

145

146

Curriculum Vitae

11.06.1974 Born in Hamburg, Germany.

1980-1993 Primary and high school in Hamburg and Braunschweig, attaining the

'Abitur' at the Jugenddorf Christophorusschule Braunschweig, Ger¬

many.

1993-1995 Study at the University of Hamburg, attaining the 'Vordiplom' in

both Mathematics and Physics; scholarship of the 'Studienstiftung des

deutschen Volkes'.

1995-1996 Studies in the MSc-Course 'Quantum Fields and Fundamental Forces'

at Imperial College (London, England), attaining the 'Imperial CollegeInternational Diploma' in Physics; supported by a grant of the German

Academic Exchange Service.

1996-1998 Study in Mathematics at ETH Zürich (supervisor Prof.Dr.E.Zehnder),attaining the diploma with distinction and the Willi Studer award.

1998-now PhD study in Mathematics (supervisor Prof.Dr.D.Salamon) and teach¬

ing assistant at ETH Zürich; research grant of the Swiss National Sci¬

ence Foundation.

147