GEOMETRIC VARIATIONAL PROBLEMS: REGULAR AND

SINGULAR BEHAVIOR

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF MATHEMATICS

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Da Rong Cheng

June 2017

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/mz112mp7761

© 2017 by Da Rong Cheng. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

ii

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Richard Schoen, Primary Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Brian White, Co-Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Or Hershkovits

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost for Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

Abstract

This thesis is devoted to the study of two of the most fundamental geometric

variational problems, namely the harmonic map problem and the minimal surface

problem.

Chapter 1 concerns the partial regularity of harmonic maps between Riemannian

manifolds that actually minimize the Dirichlet energy. In the case where both the do-

main and the target metrics are smooth, the regularity theory is rather well-developed,

and we managed to extend part of this theory to the case where the domain metric

is only bounded measurable. Specifically, we show that in this case the singular set

of an energy-minimizing map always has codimension strictly larger than two.

We then switch gears in Chapter 2 and study the existence of codimension-two

minimal submanifolds in a closed Riemannian manifold using the phase-transition

approach, which is an alternative to the classical min-max theory and has enjoyed

great success in the codimesion-one case. To be precise, our main result in this

Chapter shows that one can obtain a codimension-two stationary rectifiable varifold

as the energy concentration set of a sequence of suitably bounded critical points of

the Ginzburg-Landau functional, which was originally a model for phase transition

phenomena in superconducting materials.

In Appendix A, we consider the fundamental solution of second-order elliptic

systems in divergence form. We show that under mild assumptions on the coe�cients,

the fundamental solution can be bounded from above by the Green’s function for the

Laplacian. Such a growth estimate plays an important role in the analysis in Chapter

2, but is perhaps also interesting on its own.

iv

Acknowledgements

First and foremost, I would like to thank my advisor, Professor Richard Schoen.

This thesis would not have come into being without his inspiring guidance, constant

encouragements, and unrelenting optimism, which kept me moving forward even dur-

ing times of doubt. It has been a tremendous pleasure working with such a great

mentor, and I wish him all the best.

Secondly, I want to thank Professors Simon Brendle, Lenya Ryzhik, Leon Simon

and Brian White, for the interesting courses they taught, and for the many conver-

sations we have had. I thoroughly enjoyed learning from each of them. In addition,

I would like to thank Professor Or Hershkovits for carefully reading my dissertation

from start to finish, and providing many valuable suggestions.

Special thanks goes to the ARCS Foundation for the fellowship they awarded me

for the past two years, which gave me more freedom and flexibility than I would

otherwise have had, and allowed me to devote more time to research.

Next, I would like to express my sincere gratitude to my colleagues and friends

at Stanford and at Irvine. The past five years have been an uphill climb, but their

company made it all the more enjoyable. I’m glad to have had such wonderful peers,

and I thank each and every one of them.

Of course, I would not have come this far without the love and support of my

family, especially my parents, who have always been there for me all through my life,

and have given me much more than I can ever give back. No words can possibly

describe how much I appreciate everything they have done for me, and I will always

look up to them as examples to follow.

Finally, I am deeply grateful to the Lord our God, for the strength and wisdom

he has given me, for the many blessings he has bestowed on me, and for the light he

has shone upon my path. The road ahead is long, but I believe it is all in his hands.

”I will be glad and rejoice in thy mercy: for thou hast considered my trouble; thou

hast known my soul in adversities.”

- Psalm 31:7.

v

Contents

Abstract iv

Acknowledgements v

Chapter 1. Partial Regularity of Energy-Minimizing Maps 1

1. Introduction 1

1.1. Background and motivation 1

1.2. Notation and assumptions 3

1.3. Previous work and statements of main results 4

2. A compactness theorm for M⇤

6

2.1. �-convergence and compactness of F⇤

6

2.2. Proof of the compactness theorem 8

3. Improved partial regularity for maps in M⇤

13

3.1. Proof of the partial regularity theorem 13

3.2. The simply-connected case 15

Chapter 2. Phase Transitions and Codimension-Two Minimal Submanifolds 17

1. Introduction 17

1.1. Background and motivation 17

1.2. Previous work and statements of main results 20

1.3. Notation and terminology 24

2. Ginzburg-Landau critical points on closed manifolds 29

2.1. Preliminary estimates 29

2.2. Energy monotonicity formulae 35

2.3. The ⌘-ellipticity theorem 41

2.4. Convergence of Ginzburg-Landau solutions I: The regular part 56

2.5. Convergence of Ginzburg-Landau solutions II: The singular part 78

3. Existence of stationary rectifiable varifolds 86

3.1. The modified Ginzburg-Landau functional 87

3.2. Existence of min-max Ginzburg-Landau critical points 92

3.3. Energy estimates for Ginzburg-Landau critical points: The lower bound 93

Appendix A. The fundamental solution of divergence-form elliptic systems 95

vi

Bibliography 117

vii

CHAPTER 1

Partial Regularity of Energy-Minimizing Maps

1. Introduction

1.1. Background and motivation

One of the most important geometric variational integrals is the energy of a map

between Riemannian manifolds, whose critical points are known as harmonic maps.

In fact, the concept of a harmonic map generalizes many classical and fundamen-

tal objects in geometry. For example, when the target manifold is the real line Ror, more generally, the Euclidean space RL, the energy reduces to the usual Dirich-

let energy for functions, and harmonic maps reduce to harmonic functions. On the

other hand, if the domain manifold is the circle S1 or an interval in R, then har-

monic maps are known to parametrize geodesics in the target manifold. The theory

of harmonic maps is also closely tied with another central topic in geometric analysis,

namely the search for minimal surfaces in general ambient spaces. Since the work of

Morrey ([Mor48]), harmonic map methods have been successfully applied to estab-

lish the existence of minimal surfaces in many di↵erent situations, see, for instance

[SU81], [SY79], [CM08] and [Zho10]. More recently, harmonic maps have found

applications into a variety of other fields. For instance, we mention work by Petrides

([Pet14]), where harmonic maps were applied to study the existence and regularity

of metrics maximizing the smallest positive eigenvalue of the Laplacian on compact

surfaces.

The standard way to produce harmonic maps is of course through variational

methods, and since these are usually applied within the Sobolev space W 1,2, under-

standing the regularity of the resulting maps becomes a very important issue. Note

that this presents no di�culty when the target is an Euclidean space, since, as men-

tioned above, in this case harmonic maps are just harmonic functions and linear

elliptic theory applies. However, the situation gets complicated when the target has

curvature, which introduces a critical nonlinearity into the Euler-Lagrange equation

of the energy, preventing the use of standard estimates and making the regularity

problem much more challenging.

When the domain is a surface (i.e. two-dimensional), Morrey ([Mor48]) showed

that harmonic maps are always regular under the additional assumption that they

1

1. INTRODUCTION 2

minimize energy. Later, Sacks and Uhlenbeck ([SU81]) proved that finite-energy

harmonic maps with isolated singularities on surfaces are in fact regular. The surface

case was finally settled by Helein ([Hel91]), who established the regularity of weakly

harmonic maps defined on surfaces by constructing a special orthonormal frame on

the target and rewriting the Euler-Lagrange equation of the energy so that the critical

nonlinearity mentioned above lies in the Hardy space, yielding estimates not a↵orded

by standard theory ([CLMS93]).

The regularity properties of harmonic maps become very di↵erent when the do-

main manifold has dimension three or higher. In fact, in this case, one finds a large

number of singular harmonic maps, the simplest example being the map p(x) =

w(x/|x|) fromB3 to S2, where w is any rotation of S2. It was shown in [BCL86] that p

is not only harmonic, but actually energy-minimizing. Nonetheless, p is discontinuous

at the origin since it is homogeneous of degree zero and non-constant. Building upon

the previous example, Hardt, Lin and Poon ([HLP92]) obtained energy-minimizing

axially-symmetric harmonic maps from B3 into S2 with prescribed isolated singular-

ities on the z-axis. Recently, generalizing the above constructions, Riviere [Riv95]

was able to show the existence of harmonic maps from B3 to S2 that are everywhere

discontinuous in B3.

These examples show that, as opposed to the surface case, full regularity cannot

be expected at all when the domain has higher dimensions. Nonetheless, it is still pos-

sible to limit the singular behavior of harmonic maps under additional assumptions

such as energy-minimizing. Indeed, in the energy-minimizing case, it is known that

the examples obtained by Brezis-Coron Lieb and Hardt-Lin-Poon mentioned above

represent in a way the worst that can happen. This is a consequence of the regular-

ity theory developed by Schoen and Uhlenbeck in [SU82]. Specifically, their main

theorem reads as follows.

Theorem 1.1 (Schoen-Uhlenbeck, [SU82]). Suppose u : M ! N is an energy-

minimizing harmonic map, and suppose that the metrics on M and N are smooth.

Then u is regular away from a singular set of Hausdor↵ dimension at most n � 3,

where n is the dimension of M . Moreover, if n = 3, then the singular set consists of

isolated points.

Some remarks are in order. First of all, we note that once a harmonic map

is continuous, standard theory for elliptic systems kicks in to give higher regularity.

Therefore, in the context of harmonic map theory, regularity can very well be thought

of as continuity. Secondly, observe that the map p : B3 ! S2 defined earlier shows

that the dimension upper bound on the singular set, n�3, is optimal. Finally, while we

did not specify the regularity assumption on the metrics, an inspection of the proof in

1. INTRODUCTION 3

[SU82] shows that the domain metric has to be at least Lipschitz continuous. This

fact led us to ask the following question: How does the Schoen-Uhlenbeck theory

generalize when we relax the assumption on the domain metric? In particular, we

are interested in the case where the metric on M has only bounded measurable (L1)

coe�cients. This specific assumption is partly motivated by the De Giorgi-Nash

theory for divergence-form elliptic equations with leading coe�cients in L1, which

says that W 1,2-weak solutions of such equations are in fact Holder continuous.

Below, in Section 1.2, we introduce some notation and spell out the basic assump-

tions we will make. Then, in Section 1.3, we briefly discuss some important previous

results, before concluding the Introduction by indicating our contributions.

1.2. Notation and assumptions

Throughout the rest of this Chapter we assume that the target manifold N is

closed and equipped with a smooth Riemannian metric. For convenience, we will also

assume that N is isometrically embedded in some Euclidean space RL. Since the

results we will discuss are all local in nature, we will take the domain M to be the

unit ball B in Rn. Furthermore, we let A denote the collection of open subsets of B.

The Sobolev space W 1,2(B;N) is defined as

W 1,2(B;N) = {u 2 W 1,2(B;RL)| u(x) 2 N for a.e. x 2 B}.Note that even though the norm on W 1,2(B;N) might depend on the choice of metric

on B, the space W 1,2(B;N) itself does not, as long as it is bounded above and below

the Euclidean metric.

For a map u 2 W 1,2(B;N), its standard Dirichlet energy (with respect to the

Euclidean metric) will be denoted

E0(u,A) =

ˆA

|ru|2dx.

In addition, we introduce the energies associated to L1-metrics on B as follows: for

a positive number ⇤, we let F⇤

denote the class of funtionals E : L2(B;Rm) ⇥A ![0,+1] of the form

(1.2.1) E(u,A) =

( ´A

pggijD

j

u ·Di

u , u|A

2 W 1,2(A;Rm)

+1 , otherwise,

where g is a Riemannian metric on B with coe�cients in L1(B) satisfying

(1.2.2) ⇤�1|⇠|2 pg(x)gij(x)⇠

i

⇠j

⇤|⇠|2, for a.e. x 2 B and all ⇠ 2 Rn.

Next, we let M⇤

denote the set of maps u 2 W 1,2(B;N) that minimizes some

functional E 2 F⇤

, in the sense that for each ball Br

(x) ⇢⇢ B and each comparison

1. INTRODUCTION 4

map v 2 W 1,2(Br

(x);N) agreeing with u on @B, there holds

(1.2.3) E(u,Br

(x)) E(v, Br

(x)).

Finally, given a map u 2 M⇤

, recall that the regular set is defined to be

(1.2.4) reg(u) = {x 2 B| u is Holder continuous in a neighborhood of x} ,while the singular set is the complement of reg(u):

sing(u) = B \ reg(u).For technical reasons, we also define

(1.2.5) singE

(u) =

⇢

x 2 B| lim infr!0

r2�nE0(u,Br

(x)) � 1

2(1 + ⇤)�2✏

�

.

We remark here that if the domain metric is at least Holder continuous, then Holder

continuous harmonic maps are in fact Lipschitz (see [Sch84]), and therefore singE

(u) ⇢sing(u). On the other hand, as we shall see below, it is always true that sing(u) ⇢sing

E

(u), even for L1-metrics. Therefore, there is no need to distinguish between

singE

(u) and sing(u) when the metric on B is su�ciently regular. Nevertheless, we

do not know whether the two sets are equal in the generality we are considering,

hence the separate notations.

1.3. Previous work and statements of main results

The first step towards extending the Schoen-Uhlenbeck theory was made by Shi

([Shi96]), who generalized the ”✏-regularity” theorem in [SU82] to the case of L1-

metrics.

Theorem 1.2 (Y. Shi, [Shi96]). There exists positive numbers ✏ and ⌧ depending

only on n and ⇤ such that if u 2 M⇤

and Br

(x) ⇢⇢ B satisfies

(1.3.1) r2�nE0(u,Br

(x)) ✏.

Then

(1.3.2) (⌧r)2�nE0(u,B⌧r

(x)) 1

2r2�nE0(u,B

r

(x)).

In particular, if (1.3.1) holds, then u 2 C↵(Br/2

(x)), with ↵ depending only on n and

⇤.

Remark 1.3. Note that Theorem 1.2 implies that sing(u) ⇢ singE

(u). Moreover, as

in [SU82], a standard covering argument gives

(1.3.3) Hn�2(singE

(u)) = 0.

1. INTRODUCTION 5

Therefore, we know that Hn�2(sing(u)) = 0, which of course implies that

(1.3.4) dim(sing(u)) n� 2.

As is common for results of its type, the proof of Theorem 1.2 proceeds by con-

tradiction. Interestingly, while such arguments usually require certain compactness

theorem, Shi managed to avoid invoking any compactness result with the help of

Green’s function estimates for L1-metrics. Nonetheless, we believed that in order to

improve the bound (1.3.4) and move closer to the optimal bound (n � 3), it would

be necessary to establish a compactness result for maps in M. Despite the analytical

challenges presented by the low regularity of the metrics, we were able to obtain the

following compactness theorem.

Theorem 1.4 (C., [Che16]). Let {uk

} be a sequence of maps in M⇤

with

(1.3.5) supk

E0(uk

, Br

(x)) < +1, for each Br

(x) ⇢⇢ B.

Then, passing to a subsequence if necessary, there exists u 2 M⇤

such that

(1) uk

! u weakly in W 1,2(Br

(x);Rm) and strongly in L2(Br

(x);Rm) for each

Br

(x) ⇢⇢ B.

(2) Suppose Ek

and E are functionals in F⇤

that uk

and u minimize, respectively.

Then for each Br

(x) ⇢⇢ B, we have

(1.3.6) limk!1

Ek

(uk

, Br

(x)) = E(u,Br

(x)).

As mentioned above, the main di�culty in proving Theorem 1.4 comes from the

fact that the energies functionals in F⇤

are defined by L1-metrics, which makes it

hard to identify a limit functional E 2 F⇤

which the weak limit u would minimize.

This di�culty was overcome by focusing on the behavior of the functionals instead

of on the integrand defining them. Specifically, we make use of a compactness result

for F⇤

from the theory of �-convergence ([DM93]), a mode convergence that was

general enough to apply to our setting and at the same time strong enough for us to

get (1.3.6).

Combining Theorem 1.4 and Shi’s version of the ✏-regularity theorem (Theorem

1.2), we obtain the following improvement to (1.3.4).

Theorem 1.5 (C., [Che16]). There exists ✏ depending only on n, ⇤ and E0

such

that for all u 2 M⇤

satisfying

(1.3.7) r2�nE0(u,Br

(x)) E0

for all x 2 B1/2

, r 2 (0, 1/4),

we have Hn�2�✏(sing u \ B1/2

) = 0.

2. A COMPACTNESS THEORM FOR M⇤ 6

We note that the rather strong assumption (1.3.7) was included for lack of a

monotonicity formula for the energy, which yields energy bounds on all smaller scales

depending on the energy at a fixed scale. In other words, if we had a monotonicity

formula at our disposal, we could have replaced (1.3.7) by just a bound on the total

energy of u on B1

, which is a much weaker and more reasonable assumption. The

absence of monotonicity also prevented us from defining ”tangent maps” and applying

Federer’s dimension reduction principle, as in the original Schoen-Uhlenbeck theory,

to push the dimension bound on the singular set down to n � 3. The issue here is

that when the domain metric is only L1, standard arguments for the monotonicity

formula fail, and we have yet to find a proof or a counterexample. Nonetheless, thanks

to a universal energy bound due to Hardt, Kinderlehrer and Lin ([HKL88]), we

could remove condition (1.3.7) when N is simply-connected, and obtain the following

corollary of Theorem 1.5.

Corollary 1.6. Suppose N is simply-connected. Then there exists ✏ = ✏(⇤, N, n)

such that Hn�2�✏(sing u) = 0 for each u 2 M⇤

.

2. A compactness theorm for M⇤

2.1. �-convergence and compactness of F⇤

In this section we introduce the concept �-convergence for functionals defined

on the space L2(B;RL), which will be important to the proof of Theorem 1.4. The

notion of �-convergence was introduced by De Giorgi in the mid-1970s in order to deal

with families of functionals and study the asymptotic behavior of their minimizers.

The theory was very general and few assumptions were made on the structure of the

functionals, making it ideal for our situation. For a general introduction to the theory

of �-convergence, see the books by Dal Maso ([DM93]) or Braides ([Bra02]). We

now give the definition of �-convergence.

Definition 2.1. Let Fk

: L2(B;Rm) ! [0,+1] be functionals on L2(B;Rm). For

w 2 L2(B;Rm), we define

(2.1.1) �� lim supk!1

Fk

(w) = inf{lim supk!1

Fk

(wk

)|wk

! w in L2(B;Rm)}

(2.1.2) �� lim infk!1

Fk

(w) = inf{lim infk!1

Fk

(wk

)|wk

! w in L2}Moreover, we say that {F

k

} �-converges to F , denoted F = �� limk!1

Fk

, if

F = �� lim supk!1

Fk

= �� lim infk!1

Fk

.

Since L2(B;Rm) is a metric space, we have the following alternative characteriza-

tion for the �-limit of a sequence of functionals {Fk

}.

2. A COMPACTNESS THEORM FOR M⇤ 7

Lemma 2.2 ([DM93], Proposition 8.1). For w 2 L2(B;Rm), F (w) = ��lim supk!1 F

k

(w) =

�� lim infk!1 F

k

(w) if and only if the following two conditions hold.

(1) (lim inf-inequality) For each sequence (wk

) converging to w in L2(B;Rm),

(2.1.3) F (w) lim infk!1

Fk

(wk

).

(2) (lim sup-inequality) There exists a sequence (wk

) converging to w in L2(B;Rm)

such that

(2.1.4) lim supk!1

Fk

(wk

) F (w).

A sequence verifying (2.1.4) is called a recovery sequence.

The following compactness result is what makes �-convergence useful to us. It

shows that �-convergence is, in some sense, the right topology to use when studying

energy functionals associated with L1-metrics.

Proposition 2.3 (see also [DM93], Theorem 22.2). Let {Ek

} be a sequence in F⇤

.

Then, passing to a subsequence if necessary, there exists E 2 F⇤

such that

E(·, A) = �� limk!1

Ek

(·, A), for all A 2 A.

Proof. Suppose each Ek

is of the following form:

(2.1.5) Ek

(u,A) =

( ´A

pgk

gijk

Dj

u ·Di

u , u|A

2 W 1,2(A;Rm)

+1 , otherwise.

For each k define a functional Escal

k

on L2(B;R)⇥A ! [0,+1] by

(2.1.6) Escal

k

(w,A) =

( ´A

pgk

gijk

Dj

wDi

w , w|A

2 W 1,2(A;R)+1 , otherwise

.

Notice that given u 2 L2(B;Rm), if we write u = (u1, . . . , um) then

(2.1.7) Ek

(u,A) =m

X

l=1

Escal

k

(ul, A).

Now, since each metric gk

satisfies (1.2.2) by the definition of F⇤

, we may apply

Theorem 22.2 of [DM93] to deduce that, passing to a subsequence if necessary, we

have

�� limk!1

Escal

k

(·, A) = Escal(·, A), for all A 2 A,

where the functional Escal : L2(B;R)⇥A ! [0,+1] is of the form

(2.1.8) Escal(w,A) =

( ´A

pggijD

j

wDi

w , w|A

2 W 1,2(A;R)+1 , otherwise

,

2. A COMPACTNESS THEORM FOR M⇤ 8

with the metric g satisfying (1.2.2). Given u 2 L2(B;Rm) and A 2 A, we define

(2.1.9) E(u,A) ⌘m

X

l=1

Escal(ul, A).

Clearly E 2 F⇤

. Now we claim that E(·, A) = � � limk!1

Ek

(·, A) for all A 2 A. To

see this, we must check the lim inf- and lim sup- inequalities ( (1) and (2) in Lemma

2.2 ) for (Escal

k

(·, A)) and Escal(·, A) for each A 2 A.

For the lim inf- inequality, suppose (uk

) is a sequence converging to u in L2(B;Rm),

then for each l 2 {1, . . . ,m}, ul

k

converges to ul in L2(B;R). Since Escal

k

�-converges to

Escal, we deduce from (1) of Lemma 2.2 that, for each l, Escal(ul, A) lim infk!1

Escal

k

(ul

k

, A).

Hence

E(u,A) =m

X

l=1

Escal(ul, A) m

X

l=1

lim infk!1

Escal

k

(ul

k

, A)

lim infk!1

m

X

l=1

Escal

k

(ul

k

, A) = lim infk!1

Ek

(uk

, A),

which proves the lim inf-inequality.

For the lim sup-inequality, let u 2 L2(B;Rm). Then by (2) of Lemma 2.2, for each

l there is a sequence (ul

k

) converging to ul such that lim supk!1

Escal

k

(ul

k

, A) Escal(ul, A).

Now let uk

= (u1

k

, . . . , um

k

). Then (uk

) is a sequence converging to u in L2(B;Rm) and

lim supk!1

Ek

(uk

, A) = lim supk!1

m

X

l=1

Escal

k

(ul

k

, A) m

X

l=1

lim supk!1

Escal

k

(ul

k

, A)

m

X

l=1

Escal(ul, A) = E(u,A).

This proves the lim sup-inequality. ⇤

2.2. Proof of the compactness theorem

In this section we prove Theorem 1.4. Thus, suppose {uk

} is a sequence in M⇤

with locally uniformly bounded energy, i.e.

supk

E0(uk

, Br

(x)) < +1, for each Br

(x) ⇢⇢ B.

The first conclusion of the Theorem then follows by a standard diagonal argument,

yielding a limit map u 2 W 1,2(B;N). Hence, it remains to prove that u 2 M⇤

and

that {uk

} converges to u in the mode asserted in conclusion (2) of Theorem 1.4.

Now, for each k, by the definition of M⇤

we can find a functional Ek

2 F⇤

that’s

locally minimized by uk

. Then by Proposition 2.3, passing to a further subsequence

2. A COMPACTNESS THEORM FOR M⇤ 9

if necessary, we may assume that there is E 2 F⇤

such that

(2.2.1) E(·, A) = �� limk!1

Ek

(·, A), for each A 2 A.

Proposition 2.4. u minimizes E locally. In particular, u 2 M⇤

.

Proof. It su�ces to prove that for each ✓ 2 (1/2, 1) and each v 2 W 1,2(B✓

;N) with

u� v 2 W 1,2

0

(B✓

;Rm), we have

E(u,B✓

) E(v, B✓

).

Given such a comparison map v, we define an extension of v to B, denoted v, in the

following way.

v =

(

v , in B✓

u , in B � B✓

.

Next we fix two positive numbers � and ⌘, to be sent to zero later. Using (2.2.1) and

recalling Lemma 2.2, there exists a sequence {vk

} in W 1,2(B✓(1+⌘)

;Rm) converging to

v strongly in L2(B✓(1+⌘)

;Rm) such that

(2.2.2) lim supk!1

Ek

(vk

, B✓(1+⌘)

) E(v, B✓(1+⌘)

).

Below we will show that, based on the sequence {vk

}, we can construct a new sequence

{vk

}, still converging to v strongly in L2(B✓(1+⌘)

;Rm), such that (2.2.2) is preserved

and that vk

(x) 2 N for a.e. x. This will be done in two steps.

Step 1: Improving to L1-convergence

The construction below is inspired by [BDM80]. Since vk

converges strongly to

v in L2, we know in particular that vk

! v in measure. Hence there exists a sequence

kp

going to infinity such that

(2.2.3)

�

�

�

�

⇢

x 2 B✓(1+⌘)

| |vhk

(x)� vh(x)| > 1

p

�

�

�

�

�

<1

p, for all k � k

p

, h 2 {1, · · · ,m}.

Now we define a new sequence {wk

= (w1

k

, . . . , wm

k

)} as follows. Whenever k satisfies

kp

k < kp+1

, we define

(2.2.4) wh

k

=

8

>

<

>

:

vhk

, if |vhk

� vh| 1

p

vh + 1

p

, if vhk

> vh + 1

p

vh � 1

p

, if vhk

< vh � 1

p

.

Then wk

2 W 1,2(B✓(1+⌘)

;Rm) and kwh

k

� vhkL

1 1

p

for kp

k < kp+1

, so

(2.2.5) limk!1

kwk

� vkL

1 = 0.

2. A COMPACTNESS THEORM FOR M⇤ 10

Next, we check that {wk

} satisfies a condition analogous to (2.2.2). We start by

noticing that by (2.2.3) and the definition of wk

, we have, for kp

k < kp+1

(2.2.6)�

�

�

x 2 B✓(1+⌘)

| wh

k

(x) 6= vhk

(x)

�

� 1

p.

Moreover, again using the definition of wk

, we get

Ek

(wk

, B✓(1+⌘)

) =m

X

h=1

ˆ{wh

k

=v

h

k

}

pgk

gijk

Dj

vhk

Di

vhk

+

ˆ{wh

k

6=v

h

k

}

pgk

gijk

Dj

vhDu

vh!

Ek

(vk

, B✓(1+⌘)

) + C(⇤)m

X

h=1

ˆ{wh

k

6=v

h

k

}|dvh|2.

Letting k ! 1 and using (2.2.6), (2.2.2), we get

(2.2.7) lim supk!1

Ek

(wk

, B✓(1+⌘)

) E(v, B✓(1+⌘)

).

Step 2: Projecting back onto N

Since v(x) 2 N a.e., we infer, by (2.2.5), that

(2.2.8) d(wk

, N) converges to zero in L1(B✓(1+⌘)

).

Since N is compact, we may assume that there exists d > 0

Nd

= {x 2 Rm| d(x,N) d}is strictly contained in a tubular neighborhood of N . Let ⇡ denote the nearest-

point projection onto N defined on a neighborhood containing Nd

; then eventually

wk

= ⇡ � wk

is defined. By (2.2.7), (2.2.8) and the smoothness of ⇡ we infer that

limk!1

kwk

� vkL

1(B

✓(1+⌘);Rm

)

= 0

and that

(2.2.9) lim supk!1

Ek

(wk

, B✓(1+⌘)

) E(v, B✓(1+⌘)

).

To proceed, we need the following version of the Luckhaus lemma, originally

introduced in [Luc88]. A proof can be found in the book by L. Simon ([Sim96]).

Lemma 2.5. Let N be a compact submanifold of Rm with Nd

strictly contained in a

tubular neighborhood of N . Let L be a positive constant. Then there exists a constant

�(n, L, d) such that for all ✏ 2 (0, �), if u, v 2 W 1,2(B(1+✏)⇢

(y)� B⇢

(y);N) satisfy

⇢2�n

ˆB(1+✏)⇢(y)�B

⇢

(y)

|Du|2 + |Dv|2 L,

✏�2n⇢�n

ˆB(1+✏)⇢(y)�B

⇢

(y)

|u� v|2 �2.

2. A COMPACTNESS THEORM FOR M⇤ 11

Then there exists w 2 W 1,2(B(1+✏)⇢

(y) � B⇢

(y);N) such that w = u near @B⇢

(y),

w = v near @B(1+✏)⇢

(y) and

⇢2�n

ˆB(1+✏)⇢(y)�B

⇢

(y)

|Dw|2

C⇢2�n

ˆB(1+✏)⇢(y)�B

⇢

(y)

|Du|2 + |Dv|2 + C✏�2⇢�n

ˆB(1+✏)⇢(y)�B

⇢

(y)

|u� v|2,

where C depends on n, L and supp2N

d

|(d⇡)p

|.

To apply the lemma to our situation, we choose M large enough so that

(2.2.10) E0(uk

, B✓(1+⌘)

) + E0(wk

, B✓(1+⌘)

) < M�, 8k,and consider the following annuli

A⌘,l

= B✓(1+l

⌘

M

)

� B✓(1+(l�1)

⌘

M

)

, l = 1, 2, . . . ,M.

By (2.2.10), there exists an l such that

(2.2.11) E0(uk

, A⌘,l

) + E0(wk

, A⌘,l

) < � for infinitely many k.

Passing to a subsequence if necessary, we assume that (2.2.11) is satisfied for all k.

Next, let ⇢ = ✓(1 + (l � 1) ⌘M

) and ✏ = ⌘

2M

. Then we have

✓(1 + (l � 1)⌘

M) = ⇢ (1 + ✏)⇢ ✓(1 + l

⌘

M),

so that the annulus B(1+✏)⇢

\ B⇢

is contained in A⌘,l

, which is of course contained in

B \B✓

. Now since v = u on B \B✓

and since limk!1

kv� wk

kL

2 = limk!1

kv� uk

kL

2 = 0,

we have

(2.2.12) limk!1

ˆB(1+✏)⇢�B

⇢

|wk

� uk

|2 = 0.

Moreover, since both {Dwk

} and {Duk

} are weakly converging, it’s clear that there

is a constant L independent of k such that

⇢2�n

ˆB(1+✏)⇢�B

⇢

|Dwk

|2 + |Duk

|2 L for all k.

Moreover, by (2.2.12), as k tends to infinity, we eventually have

✏�2n⇢�n

ˆB(1+✏)⇢�B

⇢

|wk

� uk

|2 �(n, L, d)2.

Thus we can apply Lemma 2.5 with wk

, uk

in place of u, v, respectively, obtaining a

sequence {sk

} in L2(B(1+✏)⇢

� B⇢

;N) such thatˆB(1+✏)⇢�B

⇢

|Dsk

|2

2. A COMPACTNESS THEORM FOR M⇤ 12

C

ˆB(1+✏)⇢�B

⇢

|Dwk

|2 + |Duk

|2 + C✏�2⇢�2

ˆB(1+✏)⇢(y)�B

⇢

(y)

|wk

� uk

|2

C� + C✏�2⇢�2o(1).

Sending k to 1 in the above inequality, we get

(2.2.13) lim supk!1

E0(sk

, B(1+✏)⇢

� B⇢

) C�.

Now we define the maps {vk

}, to be compared with {uk

}, as follows.

(2.2.14) vk

=

8

>

<

>

:

wk

, on B⇢

sk

, on B(1+✏)⇢

� B⇢

uk

, on B � B(1+✏)⇢

.

Since �� limk!1

Ek

(·, A) = E(·, A), for all open subset A of B, by the lim inf-inequality

(2.1.3), we have

E(u,B✓

) lim infk!1

Ek

(uk

, B✓

) lim supk!1

Ek

(uk

, B✓

)

lim supk!1

Ek

(uk

, B(1+✏)⇢

)

lim supk!1

Ek

(vk

, B(1+✏)⇢

) (uk

is minimizing )

lim supk!1

�

Ek

(wk

, B⇢

) + CE0(sk

, B(1+✏)⇢

� B⇢

)�

lim supk!1

Ek

(wk

, B✓(1+⌘)

) + C� ( by (2.2.13))

E(v, B✓(1+⌘)

) + C� ( by (2.2.9)).

Since �, ⌘ > 0 are arbitrary, we send them to zero and conclude that

(2.2.15) E(u,B✓

) E(v, B✓

) = E(v, B✓

).

This completes the proof of Proposition 2.4. ⇤

Having shown that u 2 M⇤

, we next prove the second conclusion of Theorem

1.4. For each B✓

(x) ⇢⇢ B, we take the comparison map v 2 W 1,2(B✓

(x);N) in the

previous proposition to be just u itself restricted to B✓

(x). Then v would just be u

on B. Following the arguments of Proposition 2.4, we have

E(u,B✓

(x)) lim infk!1

Ek

(uk

, B✓

(x))

lim supk!1

Ek

(uk

, B✓

(x)) E(u,B✓(1+⌘)

(x)) + C�,(2.2.16)

3. IMPROVED PARTIAL REGULARITY FOR MAPS IN M⇤ 13

for all �, ⌘ > 0. Therefore, sending �, ⌘ to zero, we get

(2.2.17) lim supk!1

Ek

(uk

, B✓

(x)) E(u,B✓

(x)).

Combining this with the first inequality in (2.2.16), the second conclusion of Theorem

1.4 follows easily, and we’ve completed the proof Theorem 1.4.

3. Improved partial regularity for maps in M⇤

3.1. Proof of the partial regularity theorem

In this section we prove Theorem 1.5. As mentioned in the introduction, the

proof is by contradiction. Therefore we fix ⇤ and E0

and suppose that there exists

a sequence of maps {uk

} in M⇤

and a sequence of positive numbers ✏k

converging to

zero, such that each uk

satisfies (1.3.7) and

Hn�2�✏k(sing u

k

\ B1/2

) > 0.

Next, following [SU82] we define, for any subset A of the unit ball B,

(3.1.1) 's(A) = inf{X

i

rsi

| A ⇢ [i

Br

i

(xi

)}.

Note that 's is only an outermeasure, and not a measure in general. Nonetheless, we

recall that (see [Fed69], 2.10.2) the family {'s}s�0

detects the Hausdor↵ dimension

in the following sense.

(3.1.2) 's(A) = 0 if and only if Hs(A) = 0.

Moreover, we know from [Fed69], 2.10.19, that

lim supr!0

's(A \Br

(x))

rs� 2�s, for Hs-a.e. x 2 A.

Hence for each k we can choose xk

2 sing uk

\ B1/2

and rk

2 (0, 1/4) so that

(3.1.3)'n�2�✏

k(sing uk

\ Br

k

/2

(xk

))

rn�2�✏k

k

� cn

for some constant cn

depending only on n and not on k. Now define a sequence of

rescaled maps by letting

vk

(y) = uk

(xk

+ rk

y), y 2 B.

Then by the definition of vk

and the scaling properties of 's, (3.1.3) implies

(3.1.4) 'n�2�✏k(sing v

k

\B1/2

) � cn

.

3. IMPROVED PARTIAL REGULARITY FOR MAPS IN M⇤ 14

Since each uk

is in M⇤

, it is not hard to see that the sequence of rescaled maps {vk

}also lies in M

⇤

. Moreover, by the bound (1.3.7), we have

supk

E0(vk

, B) = supk

r2�n

k

E0(uk

, Br

k

(xk

)) E0

.

Thus by Theorem 1.4, there exists v 2 M⇤

such that the following hold.

(1) vk

! v stronlgly in L2(B✓

(x);Rm) for each B✓

(x) ⇢⇢ B.

(2) Suppose vk

and v minimize Ek

and E, respectively. Then for each B✓

(x) ⇢⇢ B,

(1.3.6) holds.

Now for each covering {Br

i

(xi

)} of singE

v \B1/2

by balls, let K = B1/2

�[i

Br

i

(xi

).

Let 2d = dist(K, sing v \ B1/2

) > 0. Then by the definition of singE

v (see (1.2.5)),

we can choose a finite covering {Bs

i

/2

(yi

)}Qi=1

of K, with yi

2 K and si

d such that

for each i,

(3.1.5) s2�n

i

E0(v, Bs

i

(yi

)) 1

2(1 + ⇤)�2✏.

Now by (1.3.6), there exists k0

such that for all k � k0

and 1 i Q,

(3.1.6) Ek

(vk

, Bs

i

(yi

)) E(v, Bs

i

(yi

)) + (2⇤)�1sn�2

i

✏.

Hence for k � k0

and 1 i Q,

E0(vk

, Bs

i

(yi

)) ⇤Ek

(vk

, Bs

i

(yi

)) (by (1.2.2))

⇤(E(v, Bs

i

(yi

)) + (2⇤)�1sn�2

i

✏) (by (3.1.6))

< ⇤(⇤E0(v, Bs

i

(yi

)) + (2⇤)�1sn�2

i

✏) (again by (1.2.2))

⇤((2⇤)�1✏+ (2⇤)�1✏)sn�2

i

(by (3.1.5))

= sn�2

i

✏.

So by Theorem 1.2, Bs

i

/2

(yi

) \ sing vk

= ;. Hence for k � k0

,

sing vk

⇢ [i

Br

i

(xi

).

Thus

(3.1.7)X

i

rn�2�✏k

i

� 'n�2�✏k(sing v

k

\ B1/2

) � cn

, for all k � k0

.

Letting k tend to infinity, we get

(3.1.8)X

i

rn�2

i

� cn

.

Since {Br

i

(xi

)} is an arbitrary covering of sing v \ B1/2

by balls, we conclude from

(3.1.1) that

'n�2(singE

v \ B1/2

) � cn

> 0.

3. IMPROVED PARTIAL REGULARITY FOR MAPS IN M⇤ 15

Hence by (3.1.2), this implies

Hn�2(singE

v \ B1/2

) > 0,

which is clearly in contradiction with (1.3.4), and the proof of Theorem 1.5 is com-

plete.

Remark 3.1. Theorem 1.5 and the structure of its proof are both inspired by a

result of White ([Whi86]) concerning area-minimizing hypersurfaces, and the ideas

can be traced back to Almgren. Of course, in that situation, a monotonicity formula

is available, and consequently one doesn’t need to assume a bound at all scaled.

3.2. The simply-connected case

In this section we assume in addition that ⇡1

(N) = {0} and prove Corollary 1.6.

We begin by stating a universal energy bound due to Hardt, Kinderlehrer and Lin.

Theorem 3.2 (Hardt-Kinderlehrer-Lin, [HKL88]). Assume N is simply-connected.

For each compact subset K of B, there exists a constant C = C(n,K,N,⇤) such that

(3.2.1) E0(u,K) C

for any u 2 M⇤

.

We now give the proof of Corollary 1.6. By Theorem 1.5 this reduces to verifying

condition (1.3.7). Given u 2 M⇤

, let E be a functional in F⇤

of which u is a local

minimizer and suppose E is given by (1.2.1) with some Riemannian metric g of class

L1.

For each Br

(x) ⇢⇢ B with x 2 B1/2

and r 2 (0, 1/4), we define

(3.2.2) ux,r

(y) = u(x+ 2ry), y 2 B.

We also define Ex,r

: L2(B;Rm)⇥A ! [0,+1] by

(3.2.3)

Ex,r

(v, A) =

( ´A

pg(x+ 2ry)gij(x+ 2ry)D

j

v(y) ·Di

v(y)dy , v|A

2 W 1,2(A;Rm)

+1 , otherwise.

Then it’s not hard to see that Ex,r

2 F⇤

and ux,r

is a local minimizer for Ex,r

. Thus

ux,r

2 M⇤

. Moreover, a straightforward computation shows that

(3.2.4) E0(u,Br

(x)) = (2r)n�2E0(ux,r

, B1/2

).

Now since ux,r

2 M⇤

, by Theorem 3.2 there is a constant C = C(n,⇤, N) such that

(3.2.5) E0(ux,r

, B1/2

) C.

3. IMPROVED PARTIAL REGULARITY FOR MAPS IN M⇤ 16

Combining (3.2.4) and (3.2.5), we get

r2�nE0(u,Br

(x)) 2n�2C.

Hence condition (1.3.7) is verified with E0

= 2n�2C and Corollary 1.6 follows imme-

diately.

CHAPTER 2

Phase Transitions and Codimension-Two Minimal

Submanifolds

1. Introduction

1.1. Background and motivation

The study of minimal submanifolds, i.e. critical points of the area functional,

has a long history that goes back to as early as the mid-eighteenth century, when

Lagrange posed what was later known as the Plateau problem, which asked for an

area-minimizer among surfaces spanning a prescribed closed curve. Since then, the ex-

istence of minimal submanifolds in various situations have remained a central topic in

geometric analysis, and e↵orts in this direction have led to some remarkable develop-

ments, most notably the birth of geometric measure theory (GMT), which provided a

general variational framework for finding critical points of the area functional. Among

the many questions one can ask concerning the existence of minimal submanifolds,

we will focus on the following one:

Given a closed, n-dimensional Riemannian manifold, does there exist a (non-

trivial) closed minimal submanifold of dimension k for each k = 1, 2, . . . , n� 1?.

The most obvious way to approach this problem would be to minimize area in

some non-trivial class, e.g. a non-zero homology class. However, this approach might

no work when the ambient manifold has no topology. For instance, it’s known that

the standard sphere doesn’t even admit stable minimal submanifolds, let alone lo-

cal minimizers ([Sim68], [LS73]). Thus, one would have to look for potentially

non-minimizing critical points via a certain ”min-max” procedure analogous to the

mountain pass theorem.

One of the earliest applications of min-max methods to the existence problem of

closed minimal submanifolds was due to Birkho↵ ([Bir17]), who showed that there

always exists a closed geodesic on a two-sphere equipped with any metric. The basic

idea is to consider di↵erent ways to ”sweep out” the sphere by a family of curves,

and then minimize the maximum length among all admissible sweep-outs. Later, in

an e↵ort to extend Birkho↵’s work to higher dimensions and codimensions, Almgren

([Alm62], [Alm65]; see also [Pit81]) developed a min-max procedure on the space

17

1. INTRODUCTION 18

of integral currents to produce stationary integral varifolds, a generalization of min-

imal submanifolds he introduced to complement the theory of integral currents and

handle the non-minimizing situation. A major consequence of Almgren’s theory is

the following general existence theorem.

Theorem 1.1 (Almgren). Let Mn be a closed Riemannian manifold. Then for k =

1, 2, · · · , n� 1, there exists a non-zero stationary integral k-varifold in M .

We note that the above theorem only establishes the existence of minimal subman-

ifolds in a weak sense, and leaves open the regularity problem. The optimal regularity

result was obtained in the codimension-one case (k = n � 1) thanks to the work of

Pitts ([Pit81]) and Schoen-Simon ([SS81]), which showed that the min-max integral

varifolds obtained via Almgren’s procedure are everywhere regular when 2 n 6

and has a singular set of dimension at most n� 7 when n � 7.

Despite being powerful and widely applicable, the Almgren-Pitts min-max theory

is also known for being highly involved. In recent years, an alternative approach to

producing minimal submanifolds has emerged that relies more on PDE methods and

avoids some of the technicalities of the Almgren-Pitts theory. Interestingly, the spark

for this development came from outside of mathematics.

As early as in the late 1970’s, people have observed that the Allen-Cahn func-

tionals ([CA77]) in material science modeling the formation of interfaces in various

two-phase materials can be viewed as approximations of the codimension-one area,

in the sense that a sequence of their critical points would converge, in a certain way,

to a minimal hypersurface, at least in a GMT sense. Roughly speaking, these func-

tionals come in families indexed by a parameter ✏ > 0 which controls the energy

distribution of the critical points. In the limit as ✏ goes to zero, the energy of the

critical points would concentrate on a codimension-one interface, which is expected

to support a minimal hypersurface. The first rigorous analysis on the behavior as ✏

goes to zero of the Allen-Cahn critical points was carried out by Modica ([Mod87]),

who showed that a sequence of minimizers would give rise to a limit interface that

solves an area-minimizing problem. Modica’s work was later considerably generalized

by Hutchinson and Tonegawa ([HT00]) to cover non-minimizing Allen-Cahn criti-

cal points, which were shown to give rise to stationary integral varifolds in the limit

under very mild assumptions. Builing upon this link, Guaraco recently gave a new

proof of the codimension-one case of Theorem 1.1 ([Gua15]) which bypassed the

Almgren-Pitts procedure on integral currents by applying min-max methods to the

Allen-Cahn functionals and exploiting the fact that for each ✏ > 0 they satisfy the

Palais-Smale compactness condition. We also mention that optimal regularity was

obtained in [Gua15] with the help of the analysis of stable Allen-Cahn critical points

1. INTRODUCTION 19

done in [Ton05], [TW12], and the general regularity theory for codimension-one

stable minimal hypersurfaces developed in [Wic14].

In light of this development, we find it natural to ask whether there exists a phase

transition approach to the codimension-two case of Theorem 1.1. The first step would

be of course to find a family of functionals that approximate the codimension-two area

in the same way the Allen-Cahn functionals approximate the codimension-one area.

Previous work suggests that a promising candidate is the Ginzburg-Landau func-

tionals, which arise from the study of phase transition in superconducting materials

([GL50]). Like the Allen-Cahn functionals, they come in families indexed by ✏ and

take the following form:

(1.1.1) E✏

(u) =

ˆM

e✏

(u) dvol; e✏

(u) =|ru|22

+(1� |u|2)2

4✏2,

where u is a complex-valued function representing the state of the material at each

point of the ambient manifoldM , with |u| ⇡ 1 corresponding to the ”superconducting

phase”, and |u| ⇡ 0 corresponding to the ”normal phase”. The critical points of E✏

satisfy the following partial di↵erential equation, often referred to as the Ginzburg-

Landau equation:

(1.1.2) ��u =1� |u|2✏2

u.

In the limit as ✏ goes to zero, it is expected that a sequence of critical points

{u✏

} would become S1-valued in most places, except for a codimension-two ”vortex

set” where the material does not demonstrate superconductivity. The hope is then

to show that the vortex set supports a codimension-two stationary integral varifold

whose area we can compute from the Ginzburg-Landau energies of the critical points.

A basic example to keep in mind is the following. Let M be the three manifold

[0, L]⇥ D with the standard product metric, where D is the unit disk in R2, and let

t and x denote the coordinates on [0, L] and D, respectively. Consider a sequence

of critical points {u✏

} where each u✏

is a minimizer of E✏

subject to the following

boundary conditions: u✏

(t, x) = x/|x| for all (t, x) 2 [0, L] ⇥ @D, and u✏

(0, x) =

u✏

(L, x) for all x 2 D. Then, the results of Bethuel, Brezis and Helein ([BBH93])

and Lin ([Lin98]) imply that {u✏

} converges smoothly away from [0, L]⇥ {0} to the

map u(t, x) = x/|x| as ✏ goes to zero. In this example, the vortex set is the line

segment ⌃ = [0, L]⇥ {0}, and it is of course a codimension-two minimal submanifold

in M . Moreover, the Ginzburg-Landau energy of u✏

has the following asymptotic

behavior as ✏ converges to zero:

(1.1.3) E✏

(u✏

) = ⇡L| log ✏|+O(1),

1. INTRODUCTION 20

where O(1) denotes a quantity that remains bounded as ✏ goes to zero. Heuristically,

the O(1)-term in (1.1.3) comes from an ✏-neighborhood of C, whereas the leading

term comes from the fact that u✏

is approaching x/|x| away from C.

In the remainder of this introduction, we briefly survey previous work on the re-

lationship between Ginzburg-Landau theory and codimension-two minimal submani-

folds, before stating our results. We then conclude by introducing the notations that

will be used in the proofs.

1.2. Previous work and statements of main results

The earliest mathematical literature on the Ginzburg-Landau functional focused

on the case where M is a simply connected domain ⌦ in R2, which should be thought

of as the cross-section of a cylindrical container as illustrated by the example in the

previous section. In their seminal work, which consists of a paper ([BBH93]) and

a book ([BBH94]), Bethuel, Brezis and Helein proved the following result on the

asymptotics of Ginzburg-Landau critical points as ✏ goes to zero.

Theorem 1.2 (Bethuel-Brezis-Helein, [BBH94]). Suppose ⌦ ✓ R2 is simply-connected

and that g : @⌦ ! S1 is a smooth function of degree d > 0. Let {u✏

} be a sequence

such that each u✏

is a solution to (1.1.2) subject to the boundary condition u✏

= g on

@⌦. Assume furthermore that

(1.2.1) E✏

(u✏

) C| log ✏|, for all ✏.

Then, there exists a collection of points {a1

, · · · , am

} in ⌦, called vortices, and a

smooth harmonic map u : ⌦ \ {a1

, · · · , am

} ! S1 such that the following hold.

(1) Passing to a subsequence if necessary, u✏

converges to u in Ck

loc

(⌦\{a1

, · · · , am

})for each k.

(2) We have the following convergence in the sense of Radon measures:

(1.2.2) lim✏!0

|log ✏|�1 e✏

(u✏

) dvol = ⇡m

X

j=1

cj

�a

j

,

where �a

j

is a point mass supported at aj

.

Remark 1.3.

(1) If ⌦ is assumed to be star-shaped, then the assumption (1.2.1) can be dropped.

(2) The configuration of the vortices is governed by the so-called ”renormalized

energy”, which depends on the distances between the vortices, the degree of u

at each aj

and the boundary map g. We refer the interested reader to Chapters

8 and 10 of the book by Bethuel, Brezis and Helein ([BBH94]).

1. INTRODUCTION 21

(3) In the case where each u✏

is a minimizer of E✏

subject to the boundary condition

g, the number of vortices is exactly d, with u having degree +1 at each vortex.

Moreover, in (1.2.2), the cj

’s are all equal to 1.

Although the two-dimensional case might not be the most interesting if one is

concerned with codimension-two minimal submanifolds, the analysis of this basic

setting paved the way for subsequent work on the higher-dimensional cases. The

case of three-dimensional domains was first studied in the minimizing setting by

Riviere ([Riv96]), whose work was later extended to general dimensions in Lin-

Riviere ([LR99]). Roughly speaking, their result says that if {u✏

} is a sequence

of Ginzburg-Landau minimizers on a convex domain ⌦ ⇢ Rn, with boundary condi-

tions {g✏

} arranged to blow up on a codimension-two submanifold S of @⌦, then, up

to a subsequence, {u✏

} converges smoothly to a S1-valued harmonic map away from

a closed set ⌃ which supports an area-minimizing (n � 2)-current T with @T = S.

Moreover, the area of T is given by

(1.2.3) M(T ) = ⇡ lim✏!0

E✏

(u✏

)

| log ✏| .

A version for the above result in the non-minimizing setting was obtained by

Lin and Riviere for three-dimensional domains ([LR01]), and by Bethuel, Brezis

and Orlandi in the higher-dimensional cases ([BBO01]). The Bethuel-Brezis-Orlandi

result can be summarized as follows.

Theorem 1.4 (Bethuel-Brezis-Orlandi, [BBO01]). Let ⌦ ⇢ Rn be simply-connected,

S a codimension-two smooth submanifold of @⌦, and {u✏

} a sequence such that each

u✏

is a solution to (1.1.2) subject to the boundary condition g✏

. The sequence g✏

is

assumed to satisfy

(1.2.4) |g✏

(x)| = 1 when dist(x, S) � ✏, and |g✏

(x)| 1 elsewhere.

Suppose furthermore that there is a constant C independent of ✏ such that

(1.2.5) |rkg(x)| C

(max{✏, dist(x, S)})k , for k = 1, 2.

On the other hand, we assume the following energy bound on {u✏

}.

(1.2.6) 0 < lim inf✏!0

E✏

(u✏

)

| log ✏| lim sup✏!0

E✏

(u✏

)

| log ✏| < 1.

Then the following hold after passing to a subsequence {u✏

k

}.(1) There exists a closed, countably (n � 2)-rectifiable set ⌃ with finite (n � 2)-

Hausdor↵ measure such that u✏

k

converges to u in Ck

loc

(⌦\⌃) for each k, where

u : ⌦ \ ⌃ ! S1 is a smooth harmonic map.

1. INTRODUCTION 22

(2) ⌃ = supp(V ), where V is a stationary rectifiable (n� 2)-varifold in the interior

of ⌦. Moreover, letting µ✏

= |log ✏|�1 e✏

(u✏

) dvol, we have, in the sense of Radon

measures,

(1.2.7) limk!1

µ✏

k

= kV k.In particular, M(V ) = lim

k!1| log ✏

k

|�1E✏

k

(u✏

k

).

Notice that all of the results discussed so far assumed the domain is a subset of Rn,

whereas we are mostly interested in the case where the domain is a manifold, either

with or without boundary. In the two-dimensional case, Theorem 1.2 was extended

to closed Riemann surfaces of any topological type by Baraket ([Bar96]). On the

other hand, one of our main results extends Theorem 1.4 to closed, simply-connected

Riemannian manifolds of dimension n � 3.

Theorem 1.5 (C.). Let (M, g) be a simply-connected compact Riemannian manifold

with @M = ; and g smooth, and let {u✏

} be a sequence such that each u✏

is a solution

to (1.1.2). Suppose furthermore that (1.2.6) holds. Then there exists a subsequence

{u✏

k

}, a countably (n�2)-rectifiable set ⌃ and a stationary rectifiable (n�2)-varifold

V supported on ⌃, such that both conclusions of Theorem 1.4 hold.

Remark 1.6. We do not know whether the varifold V obtained in Theorem 1.4 and

Theorem 1.5 has integer multiplicity a.e. on its support or not, although we expect

this to be true. We also note that in the codimension-one case, (n � 1)-varifolds

obtained out of sequences of Allen-Cahn critical points were shown to be integral

under quite general circumstances ([HT00]).

We will in fact prove the following more general Theorem, of which Theorem 1.5

is an immediate consequence.

Theorem 1.7 (C.). Let (M, g) be a compact Riemannian manifold with @M = ; and

g smooth. For each ✏ > 0, let u✏

be a solution to (1.1.2) and suppose the sequence

{u✏

} satisfies (1.2.6). Furthermore, let µ✏

= |log ✏|�1 e✏

(u✏

) dvol. Then, the following

statements hold along a subsequence {u✏

k

}:(1) There exists a closed, countably (n�2)-rectifiable set ⌃ ⇢ M with Hn�2(⌃) < 1

such that, after passing to a subsequence {µ✏

k

}, we have

(1.2.8) limk!1

µ✏

k

=| |22

dvol+⌫, in the sense of Radon measures,

where is a harmonic 1-form on all of M , and ⌫ is supported on ⌃.

1. INTRODUCTION 23

(2) For Hn�2-a.e. x 2 ⌃, the limit as r goes to zero of r2�n⌫(Br

(x)) exists. More-

over, denoting limr!0

r2�n⌫(Br

(x)) by ⇥(⌫, x) whenever the limit exists, we have

(1.2.9) 0 < ⌘/4 ⇥(⌫, x) C lim supk!1

1

|log ✏k

|ˆM

e✏

k

(uk

) dvolg

,

where ⌘ and C are both determined only by M and the metric g.

(3) The pair (⌃,⇥(⌫, x)) defines a rectifiable (n� 2)-varifold, denoted V , which is

stationary in M .

(4) In the case where M is simply-connected, we have ⌘ 0. Moreover, passing to

a further subsequence if necessary, {u✏

k

} converges smoothly on compact subsets

of M \ ⌃ to a harmonic map u : M \ ⌃ ! S1.

Remark 1.8. (1) Note that for the first three conclusions of Theorem 1.7 we do

not need M to be simply-connected.

(2) In the case where M is not simply-connected, we were not able to guarantee

that the varifold V obtained in statement (3) is non-zero, since the -term could

possibly account for all the energy.

Despite the issues mentioned in the remarks after the two Theorems above, it is

still possible to use Theorem 1.5 to prove the existence of a non-zero codimension-

two stationary rectifiable varifold in the case M is simply-connected, provided one

can produce a sequence of Ginzburg-Landau critical points verifying the assumption

(1.2.6). We have the following partial result, which handles the lower bound in (1.2.6).

Theorem 1.9 (C.). Let M be as in Theorem 1.5. Then there exists a sequence {u✏

}in W 1,2(M ;D) ( D is the unit disk in C ), with each u

✏

a solution to (1.1.2), such

that

(1.2.10) 0 < lim inf✏!0

E✏

(u✏

)

| log ✏| .

Remark 1.10. Indepedently, Stern ([Ste16]) obtained a sequence of Ginzburg-Landau

critical points satisfying both the upper bound and lower bound in (1.2.6) and thus the

existence of a codimension-two stationary rectifiable varifold in a simply-connected

closed manifold. Later, he also removed the simply-connected assumption ([Ste17]).

Below, in Section 1.3 we introduce the notation that will be used throughout this

chapter. The remainder of this chapter is then devoted to the proofs of Theorem

1.7 and Theorem 1.9, which will occupy Sections 2.1 to 2.5 and Sections 3.1 to 3.3,

respectively.

1. INTRODUCTION 24

1.3. Notation and terminology

Norms on function spaces.

We first consider RN -valued functions defined on a domain ⌦ in Rn. Given a

non-negative integer k, for a function u : ⌦ ! RN we define

|u|k;⌦

=k

X

j=0

supx2⌦

|rju(x)|.

Further specifying an exponent µ 2 (0, 1], we define

|u|k,µ;⌦

= |u|k;⌦

+ [u]k,µ;⌦

,

where the semi-norm [·]k,µ;⌦

is given by

[u]k,µ;⌦

= supx 6=y2⌦

|rku(x)�rku(y)||x� y|µ .

The space of RN -valued functions with finite | · |k;⌦

- or | · |k,µ;⌦

-norm is denoted

Ck(⌦;RN) or Ck,µ(⌦;RN), respectively.

Concerning boundary conditions, we let

Ck

c

(⌦;RN) = {u 2 Ck(⌦;RN)| supp(u) ⇢⇢ ⌦}, and

Ck

0

(⌦;RN) =�

u 2 Ck(⌦;RN)| rj (u|@⌦

) = 0 for j = 0, 1, · · · , k � 1

.

Finally, we use Ck

loc

(⌦;RN) to denote the space of functions u : ⌦ ! RN such

that u 2 Ck(⌦0;RN) for all compact subset ⌦0 of ⌦. The space Ck,µ

loc

(⌦;RN) is defined

similarly.

For a non-negative integer k and an exponent p 2 (0,1], we use k · kp;⌦

to denote

the standard Lp-norm, and the Sobolev norms are then defined by

kukk,p;⌦

=k

X

j=0

krjukp;⌦

.

The space of RN -valued functions with finite k · kk,p;⌦

-norm is, of course, denoted

W k,p(⌦;RN).

The k · kk,p;⌦

-closure of Ck

c

(⌦;RN) in W k,p(⌦;RN) is denoted with the standard

notation, W k,p

0

(⌦;RN).

For p 2 (1,1), we denote the dual space of W k,p

0

(⌦;RN) by W�k,p

0(⌦;RN), where

p0 = p/(p� 1). The norm on W�k,p

0(⌦;RN) is given by

klk�k,p

0;⌦

= supn

l(u)| u 2 W k,p

0

(⌦;RN), kukk,p;⌦

= 1o

.

To extend the Ck-norms to functions defined on a smooth compact manifold,

with or without boundary, we fix a covering {(Vi

,'i

)}Ki=1

by coordinate charts, with

1. INTRODUCTION 25

'i

: B1

⇢ Rn ! Vi

2 M , and let

|u|k;M

=K

X

i=1

|u � 'i

|k;B1 , for u : M ! RN .

The space Ck(M ;RN) will consists of functions u : M ! RN with |u|k;M

< 1. The

norms | · |k,µ;M

and k · kk,p;M

are defined in similar ways, as are the function spaces

Ck,µ(M ;RN) and W k,p(M ;RN).

In the case @M 6= ;, we write u 2 Ck

c

(M ;RN) if and only if u 2 Ck(M ;RN) and

supp(u) ⇢ int(M). The space Ck

0

(M ;RN), on the other hand, consists of functions

in Ck(M ;RN) vanishing on @M up to the (k � 1)-th order. The k · kk,p;M

-closure of

Ck

c

(M ;RN) is denoted W k,p

0

(M ;RN).

The localized spaces Ck

loc

(M ;RN), Ck,µ

loc

(M ;RN) and W k,p

loc

(M ;RN) are defined in

obvious ways.

For p 2 (1,1), the dual space of W k,p

0

(M ;RN) is denoted W�k,p

0(M ;RN), with

p0 = p/(p � 1). The norm on W�k,p

0(M ;RN), denoted k · k�k,p

0;M

, is defined exactly

as above, since the definition makes no reference to coordinates.

Finally, to extend the above definitions to sections of a vector bundle given by

⇡ : E ! M , we take a covering {(Wi

, i

)}K0i=1

such that E can be trivialized over

each Wi

by i

: ⇡�1(Wi

) ! Wi

⇥ RL, where L is the rank of E. Given a section

u : M ! E, we can then define, for instance, the norm kukk,p;M

to be

(1.3.1) kukk,p;M

=K

0X

i=1

k i

� (u|W

i

)kk,p;W

i

.

The space of sections u with kukk,p;M

< 1 would then be denoted W k,p(M ;E). It

is now obvious how to extend all the other definitions above to the vector bundle

setting.

Di↵erential forms

We are particularly interested in the vector bundle of di↵erential forms, since a

substantial portion of our analysis will be carried out there. Thus, we collect here

the relevant notations we will use. Since the focus is on introducing the notation,

we will at times be vague on the theoretical aspects. A standard reference for the

materials presented here is Volume 1 of the book by Giaquinta, Modica and Soucek

([GMS98]).

In keeping with the notations introduced in the last part, given a smooth Riemann-

ian manifold (M, g) of dimension n and a positive integer r, we use W k,p(M ;V

r T ⇤M)

to denote the space of r-forms on M with finite k ·kk,p;M

-norm. When there is no risk

1. INTRODUCTION 26

of confusion, we often shorten the notation as W k,p

r

(M), or even just W k,p

r

. Below we

first assume that @M = ;.The metric g induces a pairing h·, ·i between two covectors, given in local coordi-

nates by

h⇠, ⌘i = ⇠i1,··· ,ir⌘j1,··· ,jrg

i1j1(x) · · · girjr(x), where ⇠, ⌘ 2^

r

T ⇤x

M .

This pointwise pairing gives rise to a global inner-product between r-forms.

(⇠, ⌘) =

ˆM

h⇠(x), ⌘(x)i dvolg

, ⇠, ⌘ 2 W 1,2

r

(M).

The dual of the exterior di↵erential d with respect to this product is denoted d⇤, which

takes (r + 1)-forms to r-forms. In local coordinates, we have, for an (r + 1)-form ⇠,

(d⇤⌘)i1,··· ,ir = �gijr

i

⌘j,i1,··· ,ir .

The operators d and d⇤ are related by the fact that, for an r-form ⇠ and an (r+1)-form

⌘, there holds (d⇠, ⌘) = (⇠, d⇤⌘). The Hodge Laplacian is then defined by

�⇠ = (dd⇤ + d⇤d) ⇠,

and we see immediately that, for two r-forms ⇠, ⌘, there holdsˆM

h�⇠, ⌘i dvolg

=

ˆM

h⇠,�⌘i dvolg

=

ˆM

hd⇠, d⌘i+ hd⇤⇠, d⇤⌘i dvolg

⌘ Dg

(⇠, ⌘).

The bilinear form Dg

defined above is called the Dirichlet form. Note that we will

sometimes drop the subscript g when the metric dependence is clear from the context.

Any di↵erential form ⇠ 2 W 1,2

r

(M) satisfying

Dg

(⇠, ⌘) = 0 for all ⌘ 2 W 1,2

r

(M)

will be called a (weakly) harmonic r-form. The space of all harmonic r-forms will be

denoted Hr

. The classical Hodge Decomposition Theorem ([GMS98], Section 5.2.5,

Theorem 5) asserts that, for any w 2 L2

r

(M), there exists a unique h 2 Hr

such that

(1.3.2) w = h+ d'+ d⇤ ,

where ' 2 W 1,2

r�1

(M) and 2 W 1,2

r+1

(M) are such that d⇤' = 0 and d = 0.

In the case where @M 6= ;, for each x 2 @M we let ⌫(x) denote the unit inward

normal to @M . Then there holds the following orthogonal decomposition.

Tx

M = Tx

@M � span(⌫(x)).

1. INTRODUCTION 27

For each r, this decomposition induces a splitting onV

r Tx

M in which every element

v can be written uniquely as

v = v1

+ v2

^ ⌫(x),with v

1

2 Vr Tx

@M and v2

2 Vr�1 Tx

@M . Next, for an r-covector ⇠ 2 Vr T ⇤x

M , we

define tx

⇠ and nx

⇠ to be the unique elements inV

r T ⇤x

M satisfying, respectively, the

following two relations for all v = v1

+ v2

^ ⌫(x) 2 Vr Tx

M .

htx

⇠, vi = h⇠, v1

i,hn

x

⇠, vi = h⇠, v2

^ ⌫(x)i.According to this pointwise splitting, for an r-form ⇠ on M , we define, for x 2 @M ,

(t⇠) (x) = tx

⇠(x),(1.3.3)

(n⇠) (x) = nx

⇠(x).

Note that by the trace theorem for Sobolev spaces, both t and n extend to bounded

linear operators from W 1,p

r

(M) into Lp(@M ; (V

r T ⇤M)|@M

). We remind the reader

that (V

r T ⇤M)|@M

denotes the restriction of the bundleV

r T ⇤M to @M .

We now introduce the Sobolev space of di↵erential forms with boundary conditions

prescribed. Specifically, let us denote

W 1,p

r,t (M) =�

⇠ 2 W 1,p

r

(M)| t⇠ = 0

.

The space W 1,p

r,n is defined similarly. For p 2 (1,1), the dual space of W 1,p

r,t (M) will be

denoted W�1,p

0r

(M), where p0 = p/(p� 1). We remark here that the condition t⇠ = 0

is in fact independent of the metric on M . More precisely, it is true that t⇠ = 0 if and

only if ◆⇤⇠ = 0, where ◆ : @M ! M denotes the inclusion map. One can prove this

last statement with the help of the ”admissible charts” introduced in Section 5.2.6

of [GMS98], with the modification that the relation D'(x0, 0)en

= ⌫('(x0, 0)) in the

second line below the definition of �(0, ⇢) on page 549 only needs to hold at x0 = 0.

Letting ⇤ be the Hodge star operator on M , we mention here two basic properties

involving t, n and ⇤ (cf. [GMS98], Section 5.2.6, Proposition 1):

(1) ⇤n⇠ = t(⇤⇠), ⇤t⇠ = n(⇤⇠), t(d⇠) = d(t⇠), for an r-form ⇠.

(2) t (⇠ ^ ⌘) = t⇠ ^ t⌘, for r-forms ⇠, ⌘.

Next, given ⇠ 2 W 1,2

r

(M) and ⌘ 2 W 1,2

r+1

(M), if we define d⇤⌘ using the formula

above, the relation between d and d⇤ will now have to involve boundary terms.

(1.3.4) (d⇠, ⌘) = (⇠, d⇤⌘) +

ˆ@M

t⇠ ^ ⇤n⌘.

Regarding the second term on the right-hand side, we notice that ⇤ denotes the

Hodge star operator on M , so that ⇤n⌘ is an (n� r � 1)-form, and hence t⇠ ^ ⇤n⌘ is

1. INTRODUCTION 28

an (n� 1)-form. Moreover, by relations (1) and (2) above, we see that

t⇠ ^ ⇤n⌘ = t⇠ ^ t(⇤⌘) = t(⇠ ^ ⇤⌘),whose integral over @M is defined asˆ

@M

t(⇠ ^ ⇤⌘) ⌘ˆ@M

h⇠ ^ ⇤⌘,��!@MidHn�1,

where��!@M is the unit (n � 1)-vector field orienting @M . An important consequence

of (1.3.4) is the following identity.

(1.3.5) (d⇠, d⇤⌘) = 0,

which holds whenever ⇠ 2 W 1,2

r�1,t(M), ⌘ 2 W 1,2

r+1

(M) or ⇠ 2 W 1,2

r�1

(M), ⌘ 2 W 1,2

r+1,n(M).

For a proof, see ([GMS98], Section 5.2.6, Proposition 3(iii)).

The relationship between the Dirichlet formD and the Hodge Laplacian� = dd⇤+

d⇤d will also involve boundary terms. Specifically, for ⇠ 2 W 2,2

r

(M) and ⌘ 2 W 1,2

r

(M),

we have ([GMS98], Section 5.2.6, Proposition 2(iii)),

(1.3.6) D(⇠, ⌘) = (�⇠, ⌘) +

ˆ@M

t⌘ ^ ⇤nd⇠ + td⇤⇠ ^ ⇤n⌘.

Next, we define Hr,t to be the space of forms ⇠ 2 W 1,2

r,t (M) such that

Dg

(⇠, ⌘) = 0, for all ⌘ 2 W 1,2

r,t (M).

The boundary version of the Hodge Decomposition Theorem ([GMS98], Section

5.2.6, Theorem 4) now asserts that for each w 2 L2

r

(M), there exists a unique h 2 Hr,t

such that

w = h+ d'+ d⇤ ,

where ' 2 W 1,2

r�1,t(M) and 2 W 1,2

r+1,t(M) are such that d⇤' = 0 and d = 0.

Note that the statement is still true with the t-boundary condition replaced by the

n-boundary condition.

Classes of metrics

Most of the results we discuss below are of a local nature, and thus we will often

restrict ourselves to the case where the domain manifold is (B1

, g), where B1

⇢ Rn

is the unit ball, and g = (gij

) is a Riemannian metric given in components. The two

main classes of metrics we will consider are the following.

Definition 1.11. Fix �,⇤ > 0, we define M�,⇤

to be the class of metrics g = (gij

)

such that each gij

2 C0,1(B1

) and that the following estimates hold.

(L1) gij

(0) = �ij

and |gij

|0,1;B1 ⇤ for all i, j.

(L2) �|⇠|2 gij

(x)⇠i⇠j ��1|⇠|2, for all x 2 B1

, ⇠ 2 Rn.

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 29

Definition 1.12. Fix �,⇤ > 0 and µ 2 (0, 1], we define Mµ,�,⇤

to be the class of

metrics g = (gij

) such that each gij

2 C1,µ(B1

), with the following estimates.

(C1) gij

(0) = �ij

and |gij

|1,µ;B1 ⇤ for all i, j.

(C2) �|⇠|2 gij

(x)⇠i⇠j ��1|⇠|2, for all x 2 B1

, ⇠ 2 Rn.

Note that the t- and n-operators still make sense for these less regular metrics.

Indeed, one may start with the operators tx

and nx

, which are given pointwise and do

not depend on the regularity of the metric, and define t and n by the relation (1.3.3).

It is not hard to see that t and n map C1(M ;V

r T ⇤x

M) into C0,1(@M ; (V

r T ⇤x

M)|@M

)

when g 2 M�,⇤

, and into C1,µ(@M ; (V

r T ⇤x

M)|@M

) when g 2 Mµ,�,⇤

. One then

infers with the help of the trace theorem that both t and n extend to operators from

W 1,p

r

(M) into Lp(@M ; (V

r T ⇤M)|@M

).

2. Ginzburg-Landau critical points on closed manifolds

Below, in Sections 2.1 and 2.2 we derive some important basic properties of so-

lutions to (1.1.2). In Section 2.3 we prove the so-called ⌘-ellipticity theorem, which

resembles the standard ✏-regularity theorem appearing in many other geometric vari-

ational problems and is essential to the proof of Theorem 1.7, which we present in

Sections 2.4 and 2.5. Specifically, conclusions (1), (2) and (4) of Theorem 1.7 will be

established in Section 2.4, while the conclusion (3) will be treated in Section 2.5.

Recall that we denote the Ginzburg-Landau energy density (i.e. the integrand in

the definition of E✏

) by

e✏

(u) =|ru|22

+(1� |u|2)2

4✏2,

and we denote by µ✏

the energy distribution measure |log ✏|�1 e✏

(u) dvolg

.

2.1. Preliminary estimates

In this section, we collect some preliminary estimates on Ginzburg-Landau critical

points. We begin by recalling two standard results, stated below. The first one says

that a Ginzburg-Landau critical point in W 1,2 \ L4 is smooth, whereas the second

one gives basic pointwise estimates.

Proposition 2.1. Let u✏

2 W 1,2 \ L4(M ;C) be a critical point of E✏

, in the sense

that

(2.1.1)

ˆM

hru✏

,r⇣i+ |u✏

|2 � 1

✏2u✏

· ⇣ dvol = 0, for all ⇣ 2 W 1,2 \ L4(M ;C).

Then u✏

is in fact C1 on M , and therefore is a classical solution to

(2.1.2) ��u✏

=1� |u

✏

|2✏2

u✏

.

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 30

Proof. We will only sketch a proof as the result is more or less standard. Notice that

(2.1.1) implies that u✏

is a weak solution to

(2.1.3) �u✏

= f, f 2 L4/3(M).

Thus by the standard W 2,p-estimates (see, for example, [GT83] or [GM12]), we infer

that u✏

2 W 2,4/3(M). The result now follows from a bootstrap argument. ⇤

Remark 2.2. In the Introduction, we implicitly identified the set of critical points of

E✏

with the set of (classical) solutions to (2.1.2). This is justified by by Proposition

2.1.

Proposition 2.3. Let u✏

be a solution to (2.1.2) on M . Then we have the following

pointwise estimates.

(1) |u✏

(x)| 1, for all x 2 M .

(2) |ru✏

(x)| C✏�1 for all x 2 M , where the constant C depends only on the

metric g and the dimension n.

Proof. Again, both results are standard, and we only sketch a proof. We notice that

a straightforward computation gives

(2.1.4) ��|u

✏

|2 � 1�

= 2|ru✏

|2 + 2|u✏

|2 � 1

✏2|u✏

|2.Therefore assertion (1) follows from the maximum principle, since M is assumed to

be closed. Given (1), we observe that (2) follows from the multiplicative version of

the interior Schauder estimate. Specifically, for each geodesic ball B2r

⇢ M we have

(2.1.5) |ru✏

|0;B

r

C⇣

r�1|u✏

|0;B2r + |u

✏

|1/20;B2r

|�u✏

|1/20;B2r

⌘

,

where C depends only on the dimension of M and its Riemannian metric. We note

that the estimate (2.35) can be obtained from its additive version by a scaling argu-

ment. (See [BBH93], Lemma A.1.) ⇤

Concerning the gradient estimate in assertion (2), what we will actually use in

application is the following local version.

Proposition 2.4. Let g 2 M�,⇤

and suppose u : (B1

, g) ! D is a solution to (2.1.2),

where D is the unit disk in C. Then

|ru|0;B3/4

C✏�1,

where C depends only on n,� and ⇤.

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 31

Remark 2.5. The the proof of Proposition 2.4 will be omitted since one has only to

apply the estimate

|ru|0;B1/2

C⇣

|u|0;B1 + |u|1/2

0;B1|�u|1/2

0;B1

⌘

and note that |u|0;B1 1, |�u|

0;B1 ✏�2, and that, by standard Schauder theory,

the constant C in the above multiplicative estimate above depends only on n,� and

⇤. In fact, g only needs to be Holder continuous for Proposition 2.4 to hold, and the

proof is exactly the same.

The gradient estimate in Proposition 2.3 can be improved so that the right-hand

side involves the di↵erence between 1 and |u|2. Specifically, we have the result below.Theorem 2.6. Let u

✏

be as in Proposition 2.3. Then there holds

(2.1.6) |ru✏

(x)| C✏�2

�

1� |u✏

(x)|2� , for all x 2 M,

where C depends on the dimension n and the metric on M .

Theorem 2.6 is an immediate consequence of the local version.

Proposition 2.7. Let u✏

be as in Proposition 2.4. Then, for ✏ small enough so that

✏�2 � |Ricg

|, we have the estimate below whenever B2r

(x0

) ⇢ B1

.

(2.1.7) |ru✏

(x)|2 C�

✏�2 + r�2

� �

1� |u✏

(x)|2� , for x 2 Br

(x0

),

where C depends on n and g.

Remark 2.8. We remark that Stern ([Ste16]) has managed to obtain the same

gradient estimate.

The key to the proof of Proposition 2.7 is the following lemma.

Lemma 2.9. Let u be as in Proposition 2.7. For � > 0, define

(2.1.8) v�

=|ru

✏

|2(1� |u

✏

|2 + �).

Then, provided ✏�2 � |Ricg

|, v�

satisfies

(2.1.9) �v�

� c1

v2�

� c2

✏�2v�

,

where c1

= 2 and c2

= 4 + 2�.

Proof. For convenience we will write v for v�

and u for u✏

throughout this proof. We

first notice that

�v =�|ru|2

(1� |u|2 + �)+ 2

⌧

r|ru|2,r✓

1

(1� |u|2 + �)

◆�

+ |ru|2�✓

1

(1� |u|2 + �)

◆

⌘ I + II + III.

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 32

For I, we have the Bochner formula, which gives

(2.1.10) I =2|r2u|2

(1� |u|2 + �)+

2hru,r�ui(1� |u|2 + �)

+2Ric(ru,ru)

(1� |u|2 + �).

For II we use the chain-rule on the second term in the brackets

r✓

1

(1� |u|2 + �)

◆

= �r(1� |u|2 + �)

(1� |u|2 + �)2=

r|u|2(1� |u|2 + �)2

.

Thus we get

(2.1.11) II = 2hr|ru|2,r|u|2i(1� |u|2 + �)2

.

Finally, for III, we compute,

�

✓

1

(1� |u|2 + �)1+↵

◆

= 2|r|u|2|2

(1� |u|2 + �)3+

�|u|2(1� |u|2 + �)2

.

Thus III becomes

(2.1.12) III = 2|r|u|2|2|ru|2(1� |u|2 + �)3

+|ru|2�|u|2

(1� |u|2 + �)2.

Next, we use (2.1.2) to substitute the terms in (2.1.10), (2.1.11) and (2.1.12)

involving the Laplacian. We first consider the middle term on the right-hand side of

(2.1.10). Applying r to both sides of (2.1.2) and then taking the inner-produce with

ru gives

hr�u,rui = ✏�2(|u|2 � 1)|ru|2 + 2✏�2|u ·ru|2

= ✏�2(|u|2 � 1)|ru|2 + 1

2✏�2|r|u|2|2

Plugging this into (2.1.10), we get

(2.1.13) I =2|r2u|2

(1� |u|2 + �)+

2✏�2(|u|2 � 1)|ru|2(1� |u|2 + �)

+✏�2|r|u|2|2

(1� |u|2 + �)+

2Ric(ru,ru)

(1� |u|2 + �).

The other Laplacian term is in the second term on the right-hand side of (2.1.12).

To handle it we recall

�|u|2 = 2|ru|2 + 2hu,�ui = 2|ru|2 + 2✏�2(|u|2 � 1)|u|2.Plugging into III gives

(2.1.14) III = 2|r|u|2|2|ru|2(1� |u|2 + �)3

+ 2|ru|4

(1� |u|2 + �)2+ 2

✏�2(|u|2 � 1)|ru|2|u|2(1� |u|2 + �)2

.

Putting everything together, we get

�v =2|r2u|

(1� |u|2 + �)+

2✏�2(|u|2 � 1)|ru|2(1� |u|2 + �)

+✏�2|r|u|2|2

(1� |u|2 + �)+

2Ric(ru,ru)

(1� |u|2 + �)

(2.1.15)

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 33

+ 2hr|ru|2,r|u|2i(1� |u|2 + �)2

+ 2|r|u|2|2|ru|2(1� |u|2 + �)3

+ 2|ru|4

(1� |u|2 + �)2+ 2✏�2

(|u|2 � 1)|ru|2|u|2(1� |u|2 + �)2

.

There are 8 terms in total on the right-hand side of (2.1.15), 4 from I, 1 from II,

3 from III. Next we claim that the sum of the 1st, 5th and 6th terms is always

nonnegative by the Schwartz inequality. Specifically,

1

2(1st + 5th + 6th) � |r2u|2

(1� |u|2 + �)� 2

|ru||r|ru|||r|u|2|(1� |u|2 + �)(1/2+3/2)

+ 2|r|u|2|2|ru|2(1� |u|2 + �)3

� |r2u|2(1� |u|2 + �)

� 2|ru||r2u|||r|u|2|

(1� |u|2 + �)(1/2++3/2)

+ 2|r|u|2|2|ru|2(1� |u|2 + �)3

� |r|u|2|2|ru|2(1� |u|2 + �)3

.

Note that in the last inequality we’ve used the arithmetic-geometric mean inequality,

2ab a2 + b2, with

a =|r2u|

(1� |u|2 + �)1/2, b =

|ru||r|u|2|(1� |u|2 + �)3/2

.

Therefore we can drop the 1st, 5th and 6th terms from the right-hand side of (2.1.15).

We’ll also drop the 3rd term, which, being a square, is always nonnegative. Now we

can finish the proof of Lemma 2.9 as follows.

�v � 2✏�2(|u|2 � 1)|ru|2(1� |u|2 + �)

+2Ric(ru,ru)

(1� |u|2 + �)

+ 2|ru|4

(1� |u|2 + �)2+ 2

✏�2(|u|2 � 1)|ru|2|u|2(1� |u|2 + �)2

= 2|ru|4

(1� |u|2 + �)2+

2Ric(ru,ru)

(1� |u|2 + �)

+ 2✏�2|ru|2 (|u|2 � 1)(1� |u|2 + �) + (|u|2 � 1)|u|2

(1� |u|2 + �)2

= 2|ru|4

(1� |u|2 + �)2+

2Ric(ru,ru)

(1� |u|2 + �)+

2✏�2|ru|2(|u|2 � 1)(1 + �)

(1� |u|2 + �)2

� 2|ru|4

(1� |u|2 + �)2� 2 |Ric

g

| |ru|2(1� |u|2 + �)

� 2✏�2(1 + �)|ru|2(1� |u|2 + �)

= 2v2 � 2⇥|Ric

g

|+ ✏�2(1 + �)⇤

v.

Recalling the assumption that |Ricg

| ✏�2, we are done. ⇤

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 34

We now give the proof of Proposition 2.7 using Lemma 2.9 and a technique from

[CY75]. Again, v will denote the function v�

defined in the statement of Lemma 2.9.

Proof of Theorem 2.7. Let ' : [0,+1) ! [0, 1] be a cut-o↵ function such that '(t) =

1 for t � 1, '(t) = 0 for t 0, and that '0 c'1/2, where c is a universal constant.

For example, we can first take

(2.1.16) ⌘(t) =

(

t4(1� t)4 when 0 t 1

0 elsewhere.,

and then define ' by '(t) = A´

t

0

⌘(s)ds, with A chosen so that '(1) = 1. Then, for

t 1/2, we have,

('0(t))2 = At8(1� t)8 At8,

while, on the other hand,

'(t) =

ˆt

0

As4(1� s)4dt � A(1/2)41

4t5.

Thus we always have '0 c'1/2 where c is a universal constant.

To continue, let ⇣(x) = 1�'(d(x, x0

)/r�1), so that ⇣ is 1 on Br

(x0

) and vanished

o↵ B2r

(x0

). Then since '0 c'1/2, we have ⇣ 0 � �cr�1⇣1/2. We now compute

�(⇣v) = ⇣�v + 2hr⇣,rvi+ v�⇣(2.1.17)

� ⇣(c1

v2 � c2

✏�2v) + 2hr⇣,rvi � Cr�2v,

where we used Lemma 2.9 in the second inequality. Next, take a point x⇤ where ⇣v

attains its maximum. Then, at x⇤, we have

(2.1.18) r(⇣v)(x⇤) = 0 =) rv(x⇤) = �v(x⇤)r⇣(x⇤)

⇣(x⇤).

(We are assuming here that ⇣(x⇤) is positive, since otherwise the conclusion (2.1.7)

is vacuous.) Plugging (2.1.18) into (2.1.17), and noting that �(⇣v)(x⇤) 0, we get

that, at the point x⇤,

0 � ⇣(c1

v2 � c2

✏�2v)� 2v|r⇣|2⇣

� Cr�2v(2.1.19)

� ⇣(c1

v2 � c2

✏�2v)� Cr�2v,

where we used the fact that ⇣ 0 � �cr�1⇣1/2 in the second inequality. Now we multiply

both sides of (2.1.19) by ⇣(x⇤) to get

(2.1.20) 0 � c1

(⇣v)2 � c2

✏�2(⇣v)� Cr�2(⇣v).

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 35

Hence, we arrive at ⇣(x⇤)v(x⇤) C (✏�2 + r�2). To conclude, notice that since x⇤ is

a maximum of ⇣v, we have on Br

(x0

) that

(2.1.21) v(x) = ⇣(x)v(x) ⇣(x⇤)v(x⇤) C(✏�2 + r�2),

The proof of Proposition 2.7 is then complete upon recalling the definition of v = v�

and sending � to zero. ⇤

2.2. Energy monotonicity formulae

In this section, we show that solutions to (2.1.2) satisfy various energy monotonic-

ity formulae. For convenience, we will formulate the result in local terms. Thus, we

will fix two constants �,⇤ > 0 and work with solutions to (2.1.2) on (B1

, g), where

g = (gij

) is a metric in the class M�,⇤

, defined in Section 1.3.

Note that while we will generally deal with metrics with better regularity, Lips-

chitz metrics do come up for instance when we try to derive boundary estimates by

reflection in the case @M 6= ;. Therefore we have chosen to prove the monotonicity

formulae under this weaker regularity assumption.

Proposition 2.10. There exist constants � and C, depending only on n,� and ⇤

such that if g 2 M�,⇤

and u : (B1

, g) ! C is a solution to (2.1.2), then the following

hold for ⇢ 2 (0, 1).

@

@⇢

e�⇢⇢2�n

ˆB

⇢

(x)

|ru|22

+(1� |u|2)2

4✏2dvol

!

(2.2.1)

� 1

1 + C⇢

"

e�⇢⇢1�n

ˆB

⇢

(1� |u|2)22✏2

dvol+e�⇢⇢2�n

ˆ@B

⇢

|rr

u|2p

det(g)dHn�1

#

.

@

@⇢

e�⇢⇢2�n

ˆB

⇢

(x)

|ru|22

+

✓

1 +2

(1 + C)(n� 2)

◆

(1� |u|2)24✏2

dvol

!

(2.2.2)

� 1

1 + Ce�⇢⇢2�n

ˆ@B

⇢

1

(n� 2)

(1� |u|2)22✏2

+ |rr

u|2p

det(g)dHn�1.

where B⇢

denotes the Euclidean ball of radius ⇢, and Hn�1 is the (n� 1)-dimensional

Hausdor↵ measure.

Proof. Throughout this proof, the absolute value sign will denote norms computed

with respect to g. Also, just to avoid notational confusion, r will denote the covari-

ant derivative with respect to g, while partial di↵erentiation in coordinates will be

denoted with @. Finally, O(ra) will denote an object bounded in norm by Ara with

A depending only on n, � and ⇤.

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 36

We start by noting that, since u is a critical point of E✏

, the first variation formula

(2.1.1) gives

(2.2.3)

ˆM

hru,r⇣i+ |u|2 � 1

✏2u · ⇣ dvol = 0.

Taking a compactly supported C1-vector field X 2 C1

0

(B1

;Rn) and substituting ⇣ =

hX,rui into (2.2.3), we get, after simplifying,

(2.2.4)

ˆM

✓ |ru|22

+(1� |u|2)2

4✏2

◆

divX � hrru

X,rui�

dvol = 0.

Next we let r(x) = (x2

1

+· · ·+x2

n

)1/2 and choose X to be ⇠(r)rrr, where ⇠(r) = ⇣(r/⇢)

and ⇣ is a decreasing cut-o↵ function with ⇣(t) = 1 for t 1/2 and ⇣(t) = 0 for t � 1.

Then, by a direct computation we have

(2.2.5) rX = (r⇠0 + ⇠)rr ⌦rr + ⇠rr2r; divX = (r⇠0 + ⇠)|rr|2 + ⇠r�r.

We note here although the Hessian and the Laplacian still makes sense with C0,1-

metrics, the above identities only hold almost everywhere on B1

. Nonetheless, this

does no harm to our subsequent arguments. Plugging the identities in (2.2.5) into

(2.2.4) yieldsˆB1

e✏

(u)�

(r⇠0 + ⇠)|rr|2 +⇠r�r)(2.2.6)

� ⇥(r⇠0 + ⇠)|rr

u|2 + ⇠rr2r(ru,ru)⇤

dvol = 0.

We next compute the terms |rr|2, r�r and rrr2(ru,ru). Recalling the assumptions

on g, we have

(2.2.7) |rr|2 = gij@i

r@j

r = 1 + (gij � �ij)@i

r@j

r = 1 +O(r).

(2.2.8) rr2

i.j

r = r@ij

r + r�k

ij

@k

r =⇣

�ij

� xi

xj

r2

⌘

+O(r).

From (2.2.8) we get

(2.2.9) rr2r(ru,ru) = |ru|2 � |rr

u|2 +O(r)|ru|2,and that

(2.2.10) r�r = rgijr2

i,j

r = n� |rr|2 +O(r) = (n� 1) +O(r).

Plugging (2.2.7), (2.2.9) and (2.2.10) back into (2.2.6) and simplifying, we get

0 =

ˆB1

(1 +O(r))r⇠0e✏

(u) dvol+

ˆB1

⇠

✓

(n� 2)e✏

(u) +(1� |u|2)2

2✏2

◆

dvol(2.2.11)

�ˆB1

r⇠0|rr

u|2 dvol+ˆB1

O(r)⇠e✏

(u) dvol .

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 37

Recalling the definition of O(ra) and rearranging, we get

�ˆB1

r⇠0|rr

u|2 dvol+ˆB1

⇠(1� |u|2)2

2✏2dvol(2.2.12)

(2� n)

ˆB1

⇠e✏

(u) dvol�ˆB1

(1 + C1

r) r⇠0e✏

(u) dvol+

ˆB1

C2

r⇠e✏

(u) dvol .

Letting ⇣ increase to the characteristic function of B1

, we arrive at

⇢

ˆ@B

⇢

|rr

u|2p

det(g)dHn�1 +

ˆB

⇢

(1� |u|2)22✏2

dvol

(2.2.13)

(2� n)

ˆB

⇢

e✏

(u) + (1 + C1

⇢)⇢

ˆ@B

⇢

e✏

(u)p

det(g)dHn�1 + C2

⇢

ˆB

⇢

e✏

(u) dvol

= (1 + C1

⇢)

"

(2� n)

ˆB

⇢

e✏

(u) dvol

+ ⇢

ˆ@B

⇢

e✏

(u)p

det(g)dHn�1 +

(n� 2)C1

+ C2

1 + C1

⇢⇢

ˆB

⇢

e✏

(u) dvol

!#

.

Now we set � = (n � 2)C1

+ C2

and multiply both ends of the above inequality by

(1 + C1

⇢)�1e�⇢⇢1�n to get

1

1 + C1

⇢

"

e�⇢⇢1�n

ˆB

⇢

(1� |u|2)22✏2

dvol+e�⇢⇢2�n

ˆ@B

⇢

|rr

u|2p

det(g)dHn�1

#

(2.2.14)

e�⇢(2� n)⇢1�n

ˆB

⇢

e✏

(u) dvol

+ e�⇢⇢2�n

ˆ@B

⇢

e✏

(u)p

det(g)dHn�1 + �e�⇢⇢2�n

ˆB

⇢

e✏

(u) dvol

=@

@⇢

e�⇢⇢2�n

ˆB

⇢

e✏

(u) dvol

!

.

Thus we get (2.2.1). The inequality (2.2.2) now follows from adding the following

term to both sides of (2.2.1).

(2.2.15)@

@⇢

1

(1 + C1

)(n� 2)e�⇢⇢2�n

ˆB

⇢

(1� |u|2)22✏2

dvol

!

.

⇤

Proposition 2.10 controls the Ginzburg-Landau energy at all smaller scales in

terms of the energy at a fixed scale. We will also need a formula that has in some

sense the opposite e↵ect.

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 38

Proposition 2.11. Under the assumptions of Proposition 2.10, there exists constants

�0, C and ⇢0

, all depending only on n,� and ⇤, such that for ⇢ < ⇢0

there holds

@

@⇢

e��0⇢⇢2�n

ˆB

⇢

(x)

|ru|22

+(1� |u|2)2

4✏2dvol

!

(2.2.16)

1

1� C⇢

"

e��0⇢⇢1�n

ˆB

⇢

(1� |u|2)22✏2

dvol+e��0⇢⇢2�n

ˆ@B

⇢

|rr

u|2p

det(g)dHn�1

#

.

Proof. In the proof of Proposition 2.10, since the definition of O(ra) involves a two-

sided bound, we can use (2.2.11) to derive the following inequality instead of (2.2.12).

�ˆB1

r⇠0|rr

u|2 dvol+ˆB1

⇠(1� |u|2)2

2✏2dvol(2.2.17)

� (2� n)

ˆB1

⇠e✏

(u) dvol�ˆB1

(1� C1

r) r⇠0e✏

(u) dvol�ˆB1

C2

r⇠e✏

(u) dvol .

Again let ⇣ increase to the characteristic function of B1

, and set ⇢0

= 1/2C1

, we get,

for ⇢ < ⇢0

, that

⇢

ˆ@B

⇢

|rr

u|2p

det(g)dHn�1 +

ˆB

⇢

(1� |u|2)22✏2

dvol

(2.2.18)

� (2� n)

ˆB

⇢

e✏

(u) + (1� C1

⇢)⇢

ˆ@B

⇢

e✏

(u)p

det(g)dHn�1 � C2

⇢

ˆB

⇢

e✏

(u) dvol

= (1� C1

⇢)

"

(2� n)

ˆB

⇢

e✏

(u) dvol

+ ⇢

ˆ@B

⇢

e✏

(u)p

det(g)dHn�1 �

(n� 2)C1

+ C2

1� C1

⇢⇢

ˆB

⇢

e✏

(u) dvol

!#

.

We next set �0 = 2[(n � 2)C1

+ C2

] and multiply both ends of the above inequality

by (1� C1

⇢)�1e��0⇢⇢1�n to obtain, for ⇢ < ⇢

0

,

1

1� C1

⇢

"

e��0⇢⇢1�n

ˆB

⇢

(1� |u|2)22✏2

dvol+e��0⇢⇢2�n

ˆB

⇢

|rr

u|2p

det(g)dHn�1

#

(2.2.19)

� @

@⇢

e��0⇢⇢2�n

ˆB

⇢

e✏

(u) dvol

!

.

⇤

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 39

To end this section, we record the following two simple corollaries of Proposition

2.10. The first one can be viewed as a density estimate and will be used later in the

proof of Theorem 2.18.

Corollary 2.12. Let u and g be as in Proposition 2.10. Then for all x0

2 B1/2

and

⇢ �2/2, the following inequalities hold.

(2.2.20) ⇢2�n

ˆB

⇢

(x0)

e✏

(u) dvol C

ˆB1

e✏

(u) dvol .

(2.2.21)

ˆB

⇢

(x0)

|x� x0

|2�n

(1� |u(x)|2)2✏2

dvol C

ˆB1

e✏

(u) dvol .

In both inequalities, the constants C on the right-hand side depend only on n,�,⇤.

Proof. If x0

= 0, then (2.2.20) and (2.2.21) follow from (2.2.1) and (2.2.2), respec-

tively. If x0

6= 0, since the matrix X ⌘ (gij

(x0

)) is symmetric and positive-definite,

we have X = P tP for some P invertible. We will let A = P�1 and consider the

coordinate transform x = (y) ⌘ Ay+ x0

. From the definition of the class M�,⇤

, we

infer that

(2.2.22) �|y|2 |Ay|2 ��1|y|2.Thus, (B

�/2

) ⇢ B1/2

(x0

) ⇢ B1

and we can consider the pullback metric g ⌘ ⇤g on

B�/4

, which is related to g by

(2.2.23) (gij

(y)) = At (gij

( (y)))A.

In particular, (gij

(0)) = I by our choice of A. Furthermore, from (2.2.22), (2.2.23)

and the fact that g 2 M�,⇤

, we see that g satisfies

(1) [gij

]0,1;B

�/4 ��3/2⇤.

(2) �2|⇠|2 gij

(y)⇠i⇠j ��2|⇠|2, for all y 2 B�/4

and ⇠ 2 Rn.

Next we let u(y) = u( (y)) and notice that u : B�/2

! C solves (2.1.2) with respect

to g. Now, the fact that u is not defined on the whole unit ball B1

doesn’t a↵ect

the arguments in the proof of Proposition 2.10, and therefore (2.2.1) and (2.2.2) both

hold for u for ⇢ 2 (0,�/2) and for some constants �0 and C 0 depending only on n,�

and ⇤. From (2.2.1) we obtain

e�0⇢⇢2�n

ˆB

⇢

|ru|2g

2+

(1� |u|2)4✏2

dvolg

e�0 �4

✓

�

2

◆

2�n

ˆB

�/2

|ru|2g

2+

(1� |u|2)4✏2

dvolg

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 40

= e�0 �4

✓

�

2

◆

2�n

ˆ (B

�/2)

|ru|2g

2+

(1� |u|2)24✏2

dvolg

= e�0 �4

✓

�

2

◆

2�n

ˆB1

|ru|2g

2+

(1� |u|2)24✏2

dvolg

.

Also, notice that, since B�⇢

(x0

) ⇢ (B⇢

) by (2.2.22), we have

e�0⇢⇢2�n

ˆB

⇢

|ru|2g

2+

(1� |u|2)4✏2

dvolg

� ⇢2�n

ˆ (B

⇢

)

|ru|2g

2+

(1� |u|2)4✏2

dvolg

� ⇢2�n

ˆB

�⇢

(x0)

|ru|2g

2+

(1� |u|2)4✏2

dvolg

.

Combining the two strings of inequalities above, we arrive at

⇢2�n

ˆB

�⇢

(x0)

|ru|2g

2+

(1� |u|2)4✏2

dvolg

e�0 �4

✓

�

4

◆

2�n

ˆB1

|ru|2g

2+

(1� |u|2)24✏2

dvolg

,

for ⇢ 2 (0,�/2), and (2.2.20) is proved.

For (2.2.21), we note that, using (2.2.2) in place of (2.2.1), we getˆB

⇢

|y|2�n

(1� |u(y)|2)22✏2

dvolg

(1 + C 0)(n� 2)e�0⇢⇢2�n

ˆB

⇢

|ru|2g

2+

✓

1 +2

(1 + C)(n� 2)

◆

(1� |u|2)24✏2

dvolg

C

✓

�

2

◆

2�n

ˆB

�/2

|ru|2g

2+

(1� |u|2)24✏2

dvolg

C

ˆB1

|ru|2g

2+

(1� |u|2)24✏2

dvolg

,

for ⇢ 2 (0,�/2). Now observe that, since y = A�1(x � x0

) by definition, we get by

(2.2.22) that ˆB

⇢

|y|2�n

(1� |u(y)|2)22✏2

dvolg

�ˆ (B

⇢

)

�

�A�1(x� x0

)�

�

2�n

(1� |u(x)|2)22✏2

dvolg

�ˆB

�⇢

�

��1|x� x0

|�2�n

(1� |u(x)|2)22✏2

dvolg

=�n�2

2

ˆB

�⇢

|x� x0

|2�n

(1� |u(x)|2)2✏2

dvolg

.

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 41

Therefore we get ˆB

�⇢

|x� x0

|2�n

(1� |u(x)|2)2✏2

dvolg

C

ˆB1

|ru|2g

2+

(1� |u|2)24✏2

dvolg

,

for ⇢ 2 (0,�/2), and the proof of (2.2.21) is complete. ⇤

A second corollary of Proposition 2.10 that we mention here is a Courant-Lebesgue

type result. We will omit its proof since one can follow exactly the arguments of

Proposition II.2 in [BBO01]. Note that the inequality (II.6) needed for the proof

there would now follow from (2.2.1) instead of Lemma II.3 in [BBO01].

Proposition 2.13 ([BBO01], Proposition II.2). Under the assumptions of Proposi-

tion 2.10, there exists a radius r 2 (✏1/2, ✏1/4) such that

r3�n

ˆ@B

r

|r⌫

u|2p

det(g)dHn�1 + r2�n

ˆB

r

(1� |u|2)2✏2

dvolg

(2.2.24)

C

|log ✏|(✏1/4)2�n

ˆB

✏

1/4

e✏

(u) dvolg

C

|log ✏|ˆB1

e✏

(u) dvolg

,

where the constant C depends on the same parameters as in Proposition 2.10.

2.3. The ⌘-ellipticity theorem

This section is devoted to the demonstration of the following theorem.

Theorem 2.14. There exists constants ⌘0

, ✏0

and r0

, depending only on n and the

metric on M , such that if u is a solution to (2.1.2) with ✏ < ✏0

, satisfying

(2.3.1) r2�n

ˆB

r

(x)

e✏

(u) dvol ⌘�

�

�

log⇣ ✏

r

⌘

�

�

�

,

for some x 2 M , ⌘ < ⌘0

and r 2 [✏, r0

), then we have

(2.3.2) |u(x)| � 1� C⌘s,

where C and s are also constants depending only on n and the metric on M .

Theorem 2.14 is an immediate consequence of a local version, which we state

below as Theorem 2.17. Therefore, as in the previous section, we work with solutions

to (2.1.2) on (B1

, g). However, we will at times need slightly better regularity on g

than just Lipschitz. Therefore, unless otherwise stated, we assume that the metric g

lies in Mµ,�,⇤

, for some fixed constant �,⇤ > 0 and µ 2 (0, 1].

Before proceeding, we pause here to record two important properties of metrics in

Mµ,�,⇤

which will be used in the subsequent estimates. We will return to Theorem

2.14 after the proof of Lemma 2.16 below.

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 42

The first result we mention is a coercivity estimate coming from the absence of

harmonic forms on B1

, which implies unique solvability for the Poisson equation on

di↵erential forms.

Lemma 2.15. There exists a constant c0

= c0

(n,�,⇤, µ, r), such that

(2.3.3) Dg

(u, u) ⌘ˆB1

hdu, dui+ hd⇤g

u, d⇤g

ui dvolg

� c0

kuk21,2;B1

,

for all u 2 W 1,2

r,t (B1

) and for all g 2 Mµ,�,⇤

. In particular, for all l 2 W�1,2

r

(B1

),

there exists a unique u 2 W 1,2

r,t (B1

) such that

(2.3.4) Dg

(u, ⇣) = l(⇣), for all ⇣ 2 W 1,2

r,t (B1

),

with the estimate kuk1,2;B1 c�1

0

klk�1,2;B1.

Proof. Observe that by the Garding inequality ([GMS98], Section 5.2.6, Theorem

1), there exists constants c1

and c2

, both depending only on n,�,⇤ and r, such that

for all u 2 W 1,2

r,t (B1

) and g 2 Mµ,�,⇤

(in fact, g 2 M�,⇤

would su�ce), there holds

(2.3.5) Dg

(u, u) � c1

kruk22;B1

� c2

kuk22;B1

.

Therefore, it su�ces to prove the Lemma with k · k2;B1 in place of k · k

1,2;B1 on the

right-hand side of (2.3.3). Suppose this is false. Then for each k 2 N there would

exist gk

2 Mµ,�,⇤

and uk

2 W 1,2

r,t (B1

) such that

(2.3.6) kDg

k

(uk

, uk

) kuk

k22;B1

= 1.

Then by (2.3.5) we see that kuk

k21,2;B1

c�1

1

(c2

+ 1/k), and thus Rellich’s theorem

yields a subsequence, still denoted (uk

), and a limit u 2 W 1,2

r,t (B1

), such that

uk

! u strongly in L2 and weakly in W 1,2.

Furthermore, by the bounds imposed on the metrics in Mµ,�,⇤

, passing to a subse-

quence of (gk

) if necessary, we may assume that (gk

) converges in C1 to a limit metric

g 2 Mµ,�,⇤

.

Now, by (2.3.6) and the convergence of (uk

) and (gk

), we infer that

(2.3.7) Dg

(u, u) lim infk!1

Dg

k

(uk

, uk

) = 0.

On the other hand, the strong L2-convergence of (uk

) implies that kuk2;B1 = 1. Thus

u is a non-zero harmonic r-form on B1

with tu = 0 on @B1

, a contradiction. This

proves the estimate (2.3.3). The assertion on unique solvability now follows from

standard Hilbert space theory. ⇤

The second preliminary result we record here is a special case of the Hodge De-

composition Theorem for metrics in M�,⇤

.

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 43

Lemma 2.16. Let g 2 M�,⇤

. Then, given w 2 L2

1

(B1

), there exists a d-closed 2-form

' 2 W 1,2

2,t (B1

) and a function ⇠ 2 W 1,2

0

(B1

;R) such that

(2.3.8) w = d⇤'+ d⇠.

Proof. Since each gij

2 C0,1(B1

), it is standard that we can extend it to a larger ball

B1+s0 such that, denoting the extended function still by g

ij

, we have |gij

|0,1;B1+�0

⇤.

Now let {⇣s

}s<s0 be such that ⇣

s

(x) = s�n⇣(x/s), where ⇣ lies in C1c

(B1

) and

integrates to 1. We define a family of approximate metrics gs = (gsij

) by

(2.3.9) gsij

(x) =

ˆB

�

⇣s

(y) (gij

(x� y)� gij

(�y)) dy + �ij

, for x 2 B1

, s < s0

.

Then we see immediately that gsij

(0) = �ij

. Furthermore, there holds

gsij

(x)� gij

(x) = [(⇣s

⇤ gij

)(x)� gij

(x)]� [(⇣s

⇤ gij

)(0)� gij

(0)] ,

and hence by the regularity assumptions on g, we have

lims!0

|gsij

� gij

|0;B1 = 0,

lims!0

@gsij

(x) = @gij

(x) for Hn-a.e. x 2 B1

,

where @ denotes partial derivatives with respect to the flat metric. From (2.3.9) and

the above convergence properties, we easily see that gs 2 M�/2,⇤

for s small enough.

Since each gs is a smooth metric, Lemma 2.15 applies and we denote by us

the

unique solution in W 1,2

1,t with

(2.3.10) Dg

s(u, ⇣) = (w, ⇣) for all ⇣ 2 W 1,2

1,t (B1

).

By the smoothness of gs and the L2-theory for elliptic systems (see [Mor66] or

[GM12]) with Lipschitz leading coe�cients (note that merely g 2 Mµ,�,⇤

is not

enough for Dg

to have this property), we infer that actually us

2 W 2,2

1,t (B1

), so that

�g

su = w pointwise Hn-a.e. on B1

.

Moreover, letting 's

= dus

and ⇠s

= d⇤s

us

, where d⇤s

denotes the adjoint of d with

respect to gs, we see that d's

= 0. Furthermore, using the d-closedness of 's

, the

equation (2.3.10) and the identity (1.3.5) from Section 1.3, we see that

Dg

s('s

, ⇣) = (w, d⇤s

⇣)g

s , for all ⇣ 2 W 1,2

2,t

(B1

),(2.3.11)

Dg

s(⇠s

, ⇣) = (w, d⇣)g

s , for all ⇣ 2 W 1,2

0

(B1

;R).(2.3.12)

Note that the right-hand side of (2.3.11) defines an element in W�1,2

2

(B1

) with norm

bounded by kwk2;B1 . Thus, with the help of Lemma 2.15, we have

(2.3.13) k's

k1,2;B1 Ckwk

2;B1 ,

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 44

with C independent of s. By basic W 1,2-estimates for elliptic equations, we see that

a similar bound holds for ⇠, i.e.

(2.3.14) k⇠s

k1,2;B1 Ckwk

2;B1 .

By the above two estimates, we may pass to a subsequence {sk

} so that, as k ! 1,

the following convergence hold weakly in W 1,2 and strongly in L2.

's

k

! ' 2 W 1,2

2,t (B1

),

⇠s

k

! ⇠ 2 W 1,2

0

(B1

;R).

To conclude, we notice that for all test 1-form ⇣ 2 C1

0

(B1

;T ⇤B1

), we have

(w � d⇤'� d⇠, ⇣)g

= limk!1

(w � d⇤s

k

's

k

� d⇠s

k

, ⇣)g

s

k

= limk!1

(w ��g

s

k

u, ⇣)g

s

k

= 0,

and we are done.

We would like to point out that in the expression (w�d⇤'s

k

�d⇠s

k

, ⇣)g

s

k

above, we

are simultaneously passing to limits in the forms 's

, ⇠s

and in the metrics gs, and one

might worry that the mode of convergence we have is insu�cient for the first equality

above to hold. However, notice that the terms (w, ⇣)g

s and (d⇠s

, ⇣)g

s pose no threat

since no derivatives of gs are involved, and the metric gs itself is converging uniformly.

Moreover, by the definition of d⇤s

, we see that @gs and @' are never multiplied together

in the coordinate expression for the pairing hd⇤s

's

, ⇣ig

s and thus we can pass to the

limit with no problem. ⇤

Coming back to Theorem 2.14, we now state the local version we will prove.

In what follows, D will denote the unit disk in C.

Theorem 2.17 (see also [BBO01], Theorem 2). There exist constants ⌘0

, ✏0

, s and

C, depending on n, µ,� and ⇤, such that if g 2 Mµ,�,⇤

and u : (B1

, g) ! D solves

(2.1.2) with ✏ < ✏0

and

(2.3.15)

ˆB1

e✏

(u) dvol ⌘ |log ✏| ,

for some ⌘ < ⌘0

, then we have

(2.3.16) |u(0)| � 1� C⌘s.

The proof of Theorem 2.17 relies on the following two propositions.

Proposition 2.18 ([BBO01], Theorem 3). Let g 2 Mµ,�,⇤

and suppose u : (B1

, g) !D is a solution to (2.1.2) (not necessarily satisfying (2.3.15)), then there exists �

0

<

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 45

1/4, depending on n, µ,�, and ⇤, such that for � < �0

there holdsˆB

�

e✏

(u) dvolg

C

"

✓ˆB1

(1� |u|2)2✏2

dvolg

◆

1/3

✓ˆB1

e✏

(u) dvolg

◆

(2.3.17)

+

✓ˆB1

(1� |u|2)2✏2

dvolg

◆

2/3

+ �nˆB1

e✏

(u) dvolg

#

,

where C depends only on n, µ,� and ⇤.

Proposition 2.19 ([BBO01], Lemma III.1). Let g 2 M�,⇤

and let u : (B1

, g) ! Csolve (2.1.2). Assume in addition that (2.3.15) holds for some ⌘. Then, for � <

min{1/4, ⇢0

} (with ⇢0

given by Proposition 2.11) and ✏ < �2, there exists a radius

r0

2 (✏1/2, �) such that the following three inequalities hold.

(2.3.18) r2�n

0

ˆB

r0

(1� |u|2)22✏2

dvol K⌘ |log �| ,

(2.3.19)ˆr0

�r0

r1�n

ˆB

r

(1� |u|2)22✏2

dvol+r2�n

ˆ@B

r

|rr

u|2p

det(g)dHn�1

�

dr K⌘ |log �| ,

(2.3.20) r2�n

0

ˆB

r0

e✏

(u) dvol K(�r0

)2�n

ˆB

�r0

e✏

(u) dvol+K⌘ |log �| ,

where K is a constant depending only on n,� and ⇤.

Remark 2.20. Note that in Proposition 2.18 we required g 2 Mµ,�,⇤

and |u| 1,

while in Proposition 2.19 we only need g 2 M�,⇤

.

Proof of Theorem 2.17 assuming Propositions 2.18 and 2.19. Theorem 2.17 follows from

Propositions 2.18 and 2.19 in the exact same way Theorem 2 follows from Theorem

3 and Lemma III.1 in [BBO01]. Nevertheless, we include this argument with the

necessary modifications.

Let � < �0

be a small constant to be determined later. Then Proposition 2.19

yields a radius r0

for which (2.3.18), (2.3.19) and (2.3.20) hold. Next we will apply

Proposition 2.18 at the scale r0

. Specifically, consider the following rescaling:

(2.3.21) u(x) = u(r0

x); ✏ = ✏/r0

; gij

(x) = gij

(r0

x).

Then its easy to see that g again belongs to Mµ,�,⇤

and that u solves (2.1.2) with ✏

in place of ✏. Therefore, since � < �0

, we may apply Proposition 2.18 to get (2.3.17)

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 46

with u, ✏ and g in place of u, ✏ and g, respectively. Scaling back, we obtain

r2�n

0

ˆB

�r0

e✏

(u) dvolg

C

2

4

r2�n

0

ˆB

r0

(1� |u|2)2✏2

dvolg

!

1/3

r2�n

0

ˆB

r0

e✏

(u) dvolg

!

(2.3.22)

+

r2�n

0

ˆB

r0

(1� |u|2)2✏2

dvolg

!

2/3

+ �nr2�n

0

ˆB

r0

e✏

(u) dvolg

3

5 .

Combining the above inequality and (2.3.20) from Proposition 2.19 and recalling

(2.3.18), we get the following inequality.

r2�n

0

ˆB

r0

e✏

(u) dvol C

"

(⌘ |log �|)1/3

�2�nr2�n

0

ˆB

r0

e✏

(u) dvol

!

(2.3.23)

+ �2�n (⌘ |log �|)2/3 + �2r2�n

0

ˆB

r0

e✏

(u) dvol

#

+K⌘ |log �| .

Moving all the terms involving r2�n

0

´B

r0e✏

(u) dvol to the left-hand side, we arrive at

1� C (⌘ |log �|)1/3�n�2

� C�2!

r2�n

0

ˆB

r0

e✏

(u) dvol(2.3.24)

C�2�n (⌘ |log �|)2/3 +K⌘ |log �|C�2�n (⌘ |log �|)2/3 ,

where in going from the second to the third line we absorbed the term K⌘ |log �|. Tocontinue, we choose � = ⌘1/3n if ✏3n/2 < ⌘ (we required that �2 > ✏). Then the above

inequalities give

(2.3.25)�

1� C⌘2/3n |log ⌘|� r2�n

0

ˆB

r0

e✏

(u) dvol C⌘(n+2)/3n |log ⌘|2/3 .

Hence, if ⌘0

is chosen small enough, then we have

(2.3.26) r2�n

0

ˆB

r0

e✏

(u) dvol C⌘(n+2)/3n |log ⌘|2/3 ,

provided ✏3n/2 < ⌘ < ⌘0

. However, by the monotonicity formula (2.2.1) and the bound

(2.3.15), we easily see that (2.3.26) holds when ✏3n/2 � ⌘ as well. Therefore (2.3.26)

holds as long as ⌘ < ⌘0

. We now apply (2.2.1) again to get

(2.3.27) ✏2�n

ˆB

✏

e✏

(u) dvol e�r0r2�n

0

ˆB

r0

e✏

(u) dvol C⌘(n+2)/3n |log ⌘|2/3 .

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 47

In particular, we have

(2.3.28) ✏�n

ˆB

✏

(1� |u|2)2 dvol C⌘(n+2)/3n |log ⌘|2/3 .

We need the following result to conclude the proof.

Lemma 2.21 ([BBO01], Lemma III.3). Let u : (B1

, g) ! D be a solution to (2.1.2)

with g 2 M�,⇤

. Then, for ✏ < 1/2, we have

(2.3.29) 1� |u(0)| C

✓

✏�n

ˆB

✏

(1� |u|2)2 dvol◆

1/(n+2)

.

This Lemma is an easy consequence of the gradient estimate (2) of Proposition 2.3,

and we refer the reader to Lemma III.3 of [BBO01] for the proof, which carries over

word for word to our setting. Combining (2.3.28) and (2.3.29), we obtain (2.3.16),

and the proof is complete. ⇤

Now we shift our attention to Propositions 2.18 and 2.19. We will in fact omit the

proof of the latter, since the proof of Lemma III.1 in [BBO01] carries over directly,

with the only change being that (2.2.1) should be used in place of the monotonicity

formula (II.1) used there, and that Proposition 2.11 should be used instead of Lemma

II.2 to derive (2.3.20) from (2.3.19). Therefore, we will focus on Proposition 2.18.

Proof of Proposition 2.18. We’d like to reiterate that the arguments we give here is

a modification of the proof of Theorem 3 of [BBO01]. The main di↵erence is in the

last part of Step 2 where we estimate the 2-form '1

, to be introduced later. In what

follows, Br

and @Br

will denote Euclidean balls and Euclidean spheres, respectively.

Integrals over Br

will always be with respect to dvol =p

det(g)dx, whereas inte-

grals over @Br

will be with respect top

det(g)dHn�1, the measure induced by dvolg

.

Therefore, we will often omit the measures when writing the integrals.

The proof is rather lengthy and thus will be divided into several steps. We begin

by treating the |ru|2-term in e✏

(u), which can be decomposed as follows.

(2.3.30) |ru|2 = �1� |u|2� |ru|2 + 1

4

�

�r|u|2��2 + |u⇥ru|2,where u⇥ru ⌘ u1ru2 � u2ru1 in terms of the components of u = (u1, u2). We will

estimate these terms one by one, starting with the last term.

Step 1: Decomposition of u ⇥ ru. Let R < 3/4 be a radius which will be fixed

depending only on n,�,⇤ and µ. We first choose a radius r1

2 [R/2, R] satisfyingˆ@B

r1

|ru|2p

det(g)dHn�1 C

ˆB1

|ru|2 dvol,(2.3.31)

ˆ@B

r1

�

1� |u|2�2p

det(g)dHn�1 C

ˆB1

�

1� |u|2�2 dvol .

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 48

Note that since R will later be chosen to depend only on predetermined parameters,

we omit the R-dependences of the constants C. We will specify the choice of R at

the end of Step 2. (See (2.3.51) and the remarks preceding it.)

Next, we want to perform a Hodge decomposition on the one-form u⇥du. To that

end we let ⇠ (real-valued) be the solution to the following boundary value problem

(2.3.32)

8

>

<

>

:

�⇠ = 0 on Br1 ,

r⌫

⇠ = u⇥r⌫

u on @Br1 ,´

B

r1⇠ dvol = 0.

Since g is C1,µ and thus Lipschitz, the harmonicity of ⇠ and the fact that ⇠ integrates

to zero on Br1 imply that |r⇠|

0;B

R/4 Ckr⇠k

2;B

r1. The right-hand side can in turn

be bounded in the following wayˆB

r1

|r⇠|2 dvol =ˆ@B

r1

⇠r⌫

⇠p

det(g)dHn�1(2.3.33)

k⇠k2;@B

r1kr

⌫

⇠k2;@B

r1 Ckr⇠k

2;B

r1kruk

2;@B

r1

Ckr⇠k2;B

r1kruk

2;B1 .

Hence kr⇠k2;B

r1 Ckruk

2;B1 and we get for � < R/4 that

(2.3.34)

ˆB

�

|r⇠|2 dvol C�nˆB1

|ru|2 dvol .

Now we consider the one-form �B

r1(u⇥ du� d⇠) and computeˆ

B

r1

hu⇥ du� d⇠, d⇣i dvol =ˆ@B

r1

⇣(u⇥r⌫

u�r⌫

⇠)p

det(g)dHn�1

+

ˆB

r1

⇣d⇤(u⇥ du� d⇠) dvol = 0,

for all ⇣ 2 W 1,2

0

(B1

;R). Therefore, Lemma 2.16 yields a two-form ' 2 W 1,2

2,t (B1

) such

that

d⇤' = (u⇥ du� d⇠)�B

r1; d' = 0.(2.3.35)

k'k1,2;B1 C(kduk

2;B

r1+ kd⇠k

2;B

r1) Ckruk

2;B1 .

Therefore, on Br1 we have

(2.3.36) u⇥ du = d⇤'+ d⇠.

The second term on the right-hand side is already estimated by (2.3.34). We next

deal with the first term.

Step 2: Estimates for d⇤'.

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 49

This is the longest step in the proof. Note that although ' satisfies �' = dd⇤' =

du ⇥ du, this is not a very useful equation since the right-hand side is di�cult to

control. To remedy this, we follow [BBO01] and introduce

↵(x) =

(

f(|u(x)|)2 in Br1

1 elsewhere.,

where the function f satisfies |f 0| 4 and is given as follows for some � < 1/4 to be

determined later.

f(t) =

(

1

t

if t � 1� �

1 if t 1� 2�.

Later in the proof, we will need to use the simple observation that 1�↵ 4�, which

can easily be verified from the definition.

Notice that if we multiply equation (2.3.36) by ↵ and apply d to the result, then

we get, on Br1 ,

d(↵d⇤') = d(↵u⇥ du)� d(↵d⇠)(2.3.37)

= d (f(|u|)u)⇥ d (f(|u|)u)� d(↵d⇠).

Hence, on Br1 , we have

(2.3.38) �' = d (f(|u|)u)⇥ d (f(|u|)u)� d(↵d⇠) + d((1� ↵)d⇤').

The second and third terms on the right-hand side will turn out to be relatively easy

to handle. Further more, the first term on the right-hand side is now easier to deal

with than du⇥ du. In fact, we have

Claim. For some constant depending only on n, µ,�,⇤, we have

(2.3.39) |d(↵u⇥ du)| C(1� |u|2)2

�2✏2, pointwise on B

3/4

.

Proof of the claim. Observe that when |u(x)| � 1� �, we have

(2.3.40) d (f(|u|)u)⇥ d (f(|u|)u) = d

✓

u

|u|◆

⇥ d

✓

u

|u|◆

= 0,

where the last equality holds because u maps into C. Therefore, we obtain the

following bound on B3/4

.

|d(↵u⇥ du)| = |d (f(|u|)u)⇥ d (f(|u|)u)|(2.3.41)

�{|u|1��}|du|2 C(1� |u|2)2

�2✏2,

where in the last inequality we used Proposition 2.4 and the fact that 1 � |u|2 � �

whenever |u(x)| 1 � �. We shall see below that the right-hand side of (2.3.39) is

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 50

indeed a more tractible term than du ⇥ du, because it appears in the monotonicity

formula (2.2.1). ⇤

Motivated by the Claim, below we will use (2.3.38) instead of �' = du ⇥ du to

derive estimates for '. To proceed, we will need a version of (2.3.38) that holds on

B1

rather than just on Br1 . For that purpose, we let ⇣ be a two-form in W 1,2

2,t (B1

)

and compute

(2.3.42)

ˆB1

h↵d⇤', d⇤⇣i =ˆB

r1

h↵u⇥ du, d⇤⇣i �ˆB

r1

h↵d⇠, d⇤⇣i = (I) + (II)

For (I), we have

(I) =

ˆ@B

r1

t↵u⇥ du ^ t ⇤ ⇣ +ˆB

r1

hd(↵u⇥ du), ⇣i,

While for (II), we have

(II) =

ˆB

r1

h(1� ↵)d⇠, d⇤⇣i �ˆB

r1

hd⇠, d⇤⇣i

=

ˆB

r1

h(1� ↵)d⇠, d⇤⇣i �ˆ@B

r1

td⇠ ^ t ⇤ ⇣.

Therefore, putting everything together, ' satisfies

ˆB1

hd⇤', d⇤⇣i =ˆB

r1

hd(↵u⇥ du), ⇣i+ˆ@B

r1

t↵u⇥ du ^ t ⇤ ⇣(2.3.43)

+

ˆB

r1

h(1� ↵)d⇠, d⇤⇣i �ˆB

r1

hd⇠, d⇤⇣i+ˆB1

h(1� ↵)d⇤', d⇤⇣i

⌘ w1

(⇣) + w2

(⇣)

+ w3

(⇣) + w4

(⇣) + w5

(⇣),

for all ⇣ 2 W 1,2

2,t (B1

), where the five wi

’s are bounded linear functionals on W 1,2

2,t (B1

)

defined by the five terms in the two lines above them, respectively. Given (2.3.43),

we can invoke Lemma 2.15 and decompose ' into '1

+ · · ·'5

, with each 'i

being the

unique solution in W 1,2

2,t (B1

) to

(2.3.44) D('i

, ⇣) = wi

(⇣), for all ⇣ 2 W 1,2

2,t (B1

).

We first handle '3

and '5

as they are the easiest to deal with. For instance, we

can take '3

itself to be a test form in (2.3.44) for i = 3, obtaining

D('3

,'3

) = w3

('3

) =

ˆB

r1

h(1� ↵)d⇠, d⇤'3

i

k(1� ↵)r⇠k2;B

r1kd⇤'

3

k2;B1 C�kruk

2;B1kd⇤'3

k2;B1 ,

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 51

where in the last inequality we used the estimate (2.3.33) and the fact that |1�↵| 4�. Hence, since D controls the W 1,2-norm by Lemma 2.15, we get

(2.3.45) k'3

k1,2;B1 C�kruk

2;B1 .

A similar argument shows that

(2.3.46) k'5

k1,2;B1 C�kruk

2;B1 ,

only that this time we use the W 1,2-estimate for ' in (2.3.35) in place of (2.3.33).

Next, we turn to '2

and '4

. For '2

, we have

D('2

,'2

) = w2

('2

) =

ˆ@B

r1

t↵u⇥ du ^ t ⇤ '2

kruk2;@B

r1k'

2

k2;@B

r1 Ckruk

2;B1k'2

k1,2;B

r1.

Note that the last inequality follows from our choice of r1

and the trace inequality

for W 1,2

2,T

(Br1). We therefore obtain

(2.3.47) k'2

k1,2;B1 Ckruk

2;B1 .

To proceed, we notice that '2

is harmonic in B1

since D('2

, ⇣) = w2

(⇣) = 0 for all ⇣

with compact support in Br1 . Thus, since g is C1,µ, the interior Schauder estimates

give |'2

|0;B

R/4+ |r'

2

|0;B

R/4 Ck'

2

k1,2;B1 , and hence, for � < R/4,

(2.3.48) k'2

k21,2;B

�

C�nk'2

k21,2;B1

C�nkruk22;B1

.

The estimate for '4

follows a similar approach. First, we have

D('4

,'4

) = �ˆB

r1

hd⇠, d⇤'4

i kr⇠k2;B

r1k'

4

k1,2;B

r1,

and therefore the following W 1,2-estimate

k'4

k1,2;B1 Ckr⇠k

2;B

r1 Ckruk

2;B1 .

Then we notice that w4

(⇣) can be written alternatively as a boundary term, namely

w4

(⇣) =´@B

r1td⇠ ^ t ⇤ ⇣. Therefore, like '

2

, '4

is also harmonic in Br1 , and hence

(2.3.49) k'4

k21,2;B

�

C�nkruk22;B1

.

Finally we estimate '1

. Here we will need to use Corollary A.9 from Appendix A,

which gives an estimate on the fundamental solution of elliptic systems in divergence

form. To see the relevance of this result to our situation, note that if we fix a basis

(e↵

= dxk

↵ ^ dxl

↵)1↵n(n�1)/2

ofV

2 Rn and identify 2-forms on B1

with Rn(n�1)/2-

valued functions, then the Hodge Laplacian � with respect to a metric h is an elliptic

operator of the form described in (A.1), with N = n(n� 1)/2. Moreover, the leading

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 52

coe�cients of � are given by

(2.3.50) Aij

↵�

=p

det(h)hijhk

↵

k

�hl

↵

l

� ,

and the remaining coe�cients are given in terms of the components (hij) and the

Christo↵el symbols of h.

Therefore, if h 2 M�,⇤

, then the Hodge Laplacian satisfies the assumptions (H1)

to (H3) in Appendix A, with µ = 1, � = �(n,�) and ⇤ = ⇤(n,�,⇤). If we further

require that h 2 Mµ,�,⇤

, then assumption (H4) is also verified, with c0

equal to the

constant c0

(n,�,⇤, µ, 2) given by Lemma 2.15.

Returning to the proof of Theorem 2.17, since g is assumed to be in Mµ,�,⇤

, we

see from the above discussion that the Hodge Laplacian � with respect to g satisfies

assumptions (H1) to (H4) in Appendix A, and thus all the results developed there

apply to �. In particular, Corollary A.9 holds for the fundamental solution of �. We

now fix the radius R introduced at the start of Step 1 to be

(2.3.51) R = min{R2

(n,N, c0

, �, ⇤, µ),�2

6, 1/2}.

where R2

is given by Corollary A.9 with N = n(n � 1)/2 and c0

, �, ⇤, µ as in the

previous paragraph. Note that this choice of R indeed depends only on n,�,⇤ and µ.

We proceed to estimate '1

. First, notice that by (2.3.35) and the estimates for

'2

, · · ·'5

, we have

(2.3.52) k'1

k1,2;B1 k'k

1,2;B1 +5

X

i=2

k'i

k1,2;B1 Ckruk

2;B1 .

Next, since d(↵u ⇥ du)�B

r12 L1(B

1

;RN), by Theorem A.2 there exists a unique '

in W 1,2

0

(B1

;RN) satisfying

(2.3.53) D(', ⇣) = w1

(⇣) =

ˆB

r1

hd(↵u⇥ du), ⇣i, for all ⇣ 2 W 1,2

0

(B1

;RN).

By Proposition A.17(3) and the remark after its proof, denoting by (G↵�

) the fun-

damental solution of �, we see that ' can be written as the following integral for

Hn-a.e. x 2 B1

.

(2.3.54) '�

(x) =

ˆB

r1

(w1

)↵

(y)G↵�

(x, y)dy,

where, by abuse of notation, we’ve denoted d(↵u ⇥ du) also as w1

. Now we invoke

Corollary A.9 and recall (2.3.39) to obtain, for x 2 Br1 ⇢ B

R2 ,

|'(x)�

| ˆB

r1

|(w1

)↵

(y)||G↵�

(x, y)|dy(2.3.55)

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 53

C��2

ˆB

r1

|x� y|2�n

(1� |u|2)2✏2

dy

C��2

ˆB

r1

|x� y|2�n

(1� |u|2)2✏2

dvolg

Note that, in applying the estimate (A.13), we dropped the second term on the right-

hand side since it is dominated by |x�y|2�n. Also, in the last inequality, we estimated

the Euclidean volume form by C�

dvolg

and absorbed the constant.

Now, the last term in (2.3.55) can estimated using inequality (2.2.21) from Corol-

lary 2.12. Specifically, since we’ve required r1

R �2/6, (2.2.21) can be used to

show that

C��2

ˆB

r1

|x� y|2�n

(1� |u|2)2✏2

dvol C��2

ˆB3r1 (x)

|x� y|2�n

(1� |u|2)2✏2

dvol

C��2

ˆB1

e✏

(u) dvol .

Putting this together with (2.3.55) yields

(2.3.56) |'|0;B

r1 C��2

ˆB1

e✏

(u) dvol .

Pluggin in ' as a test function in (2.3.53), and using (2.3.41), (2.3.56) and (2.3.3),

we get

c0

k'k21,2;B1

D(', ') ˆB

r1

|'||w1

| dvol(2.3.57)

|'|0;B

r1

ˆB

r1

|w1

| dvol

C��4

✓ˆB1

e✏

(u) dvol

◆✓ˆB1

(1� |u|2)2✏2

◆

.

Next we note that by (2.3.53) and (2.3.44) with i = 1, the di↵erence '1

� '

satisfies

(2.3.58) D('1

� ', ⇣) = 0, for all ⇣ 2 W 1,2

0

(B1

;RN).

Therefore the interior Schauder estimates yield

|'1

� '|21,0;B1/2

Ck'1

� 'k21,2;B1

Ckruk22;B1

+ C��4

✓ˆB1

e✏

(u) dvol

◆✓ˆB1

(1� |u|2)2✏2

◆

,

where we used (2.3.52) and (2.3.57) in the last inequality. From the above C1-estimate

we infer that, for � < R/4,

k'1

� 'k21,2;B

�

C�nkruk22;B1

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 54

+ C�n��4

✓ˆB1

e✏

(u) dvol

◆✓ˆB1

(1� |u|2)2✏2

◆

,

and hence, whenever � < R/4,

k'1

k21,2;B

�

2k'1

� 'k21,2;B

�

+ 2k'k21,2;B1

(2.3.59)

C�nkruk22;B1

+ C(1 + �n)��4

✓ˆB1

e✏

(u) dvol

◆✓ˆB1

(1� |u|2)2✏2

◆

.

Putting together the estimates we derived for the 'i

’s, we arrive at

k'k21,2;B

�

C(�2 + �n)kruk22;B1

(2.3.60)

+ C��4

✓ˆB1

e✏

(u) dvol

◆✓ˆB1

(1� |u|2)2✏2

◆

.

From this and (2.3.34) and recalling the decomposition (2.3.36), we obtain the fol-

lowing estimate on u⇥ du which holds whenever � < R/4.

ku⇥ duk21,2;B

�

C(�2 + �n)kruk22;B1

(2.3.61)

+ C��4

✓ˆB1

e✏

(u) dvol

◆✓ˆB1

(1� |u|2)2✏2

◆

.

Step 3: Estimates for |r|u|2|2 and (1� |u|2)|ru|2.The estimates in this step are done exactly the same as in [BBO01]. We include

the simple arguments for convenience. We recall (2.1.4):

(2.3.62) ��|u|2 � 1

�

= 2|ru|2 + 2|u|2 � 1

✏2|u|2.

Multiplying by (1� |u|2) and integrating by parts over Br1 , we obtainˆ

B

r1

�

�r|u|2��2 + 2(1� |u|2)2|u|2✏2

=2

ˆB

r1

(1� |u|2)|ru|2 +ˆ@B

r1

(1� |u|2)r⌫

|u|2

⌘ I + II.

We split (I) into two integrals according to whether 1� |u(x)|2 > �2, and notice that,ˆ{1�|u|2�2}

(1� |u|2)|ru|2 �2

ˆB1

|ru|2ˆ{1�|u|2>�2}

(1� |u|2)|ru|2 C��2

ˆB1

(1� |u|2)2✏2

,

where we used the estimate |ru| C✏�1 in the second inequality. Combining the

two inequalities above gives

(2.3.63) |(I)| �2

ˆB1

|ru|2 + C��2

ˆB1

(1� |u|2)2✏2

.

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 55

As for (II), we recall the choice of r1

(see (2.3.31)) and use Holder’s inequality to get

|(II)| C✏

✓ˆB1

(1� |u|2)2✏2

◆

1/2

✓ˆB1

|ru|2◆

1/2

�2

ˆB1

|ru|2 + C��2

ˆB1

(1� |u|2)2✏2

,

where in the second step we dropped ✏ and used Young’s inequality. Adding the two

above estimates together, we get

(2.3.64)

ˆB

r1

�

�r|u|2��2 �2

ˆB1

|ru|2 + C��2

ˆB1

(1� |u|2)2✏2

.

Step 4: Conclusion.

Combining the estimates (2.3.61), (2.3.64) and (2.3.63), we infer that, for � < r1ˆ

B

�

e✏

(u) C(�2 + �n)kruk22;B1

+ C��2

ˆB1

(1� |u|2)2✏2

+ C��4

✓ˆB1

e✏

(u) dvol

◆✓ˆB1

(1� |u|2)2✏2

◆

.

Now we distinguish two cases. If

p✏

⌘ˆB1

(1� |u|2)2✏2

(1/8)6,

then we can choose � = p1/6✏

and get (2.3.17). On the other hand, if p✏

� (1/8)6,

then (2.3.17) obviously holds with C = (1/8)2. Hence we have completed the proof

of Proposition 2.18. ⇤

To conclude this section, we remark that having obtained Theorem 2.17 and hence

Theorem 2.14, we can now follow the proof of Proposition 1 in (BBO) to obtain the

following result, which will be used later in the proof of Theorem 1.7.

Proposition 2.22. Let u : M ! C be a solution to (2.1.2). For � 2 (1/2, 1), let

(2.3.65) A�

⌘ {x 2 M | |u| 1� �}.Then we have

(2.3.66)

ˆA

�

(1� |u|2)2✏2

dvol K

⌘�

✓

1

|log ✏|ˆM

e✏

(u) dvol

◆

2

,

where K depends only on M and its metric g, and ⌘�

= min{⌘0

, (�/2C)1/s}, withC, s as in Theorem 2.14.

Remark 2.23. We shall omit the proof of Proposition 2.22 as it is almost identical

to that of Proposition 1 in [BBO01]. We note that the Proposition II.2 used there

should be replaced by our Proposition 2.13, and Theorem 3 there should be replaced

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 56

by our Theorem 2.14. Also, one can ignore the set Kµ

defined on page 476 since there

is no boundary to worry about in our case. Finally, in the conclusion of Step 2 in the

proof there, the constant C�1/↵ should be replaced by ⌘�

, defined above.

2.4. Convergence of Ginzburg-Landau solutions I: The regular part

Sections 2.4 and 2.5 are devoted to the proof of Theorem 1.7. In this section,

we treat assertions (1), (2) and (4), whereas assertion (3) will be proved in the next

section. Let us now recall the setting of Theorem 1.7 and the conclusions (1) and (2).

We will address conclusion (4) separately at the end of this section.

The Riemannian manifold (M, g) is assumed to be compact, with @M = ; and g

smooth. For each ✏ > 0 we take a solution u✏

to (2.1.2) and suppose that the energy

bounds (1.2.6) hold. Also, we introduce the Radon measures µ✏

= |log ✏|�1 e✏

(u✏

) dvol.

Then what we aim to establish in the present section are the following two statements.

(1) There exists a closed, countably (n� 2)-rectifiable set ⌃ ⇢ M with Hn�2(⌃) <

1 such that, after passing to a subsequence {µ✏

k

}, we have

(2.4.1) limk!1

µ✏

k

=| |22

dvol+⌫, in the sense of Radon measures,

where is a harmonic 1-form on all of M , and ⌫ is supported on ⌃.

(2) For Hn�2-a.e. x 2 ⌃, the limit as r goes to zero of r2�n⌫(Br

(x)) exists. More-

over, denoting limr!0

r2�n⌫(Br

(x)) by ⇥(⌫, x) whenever the limit exists, we have

(2.4.2) 0 < ⌘/4 ⇥(⌫, x) C lim supk!1

1

|log ✏k

|ˆM

e✏

k

(uk

) dvolg

,

where ⌘ and C are both determined only by M and the metric g.

As usual, we discuss the necessary ingredients before giving the actual proof.

These are Proposition 2.24, Corollary 2.26 and Proposition 2.27 below. On the other

hand, during the proof we will make use of a few technical devices which are perhaps

not totally standard. An account on these tools can be found after the proof.

Also, as in the previous sections we will localize and work with solutions to (2.1.2)

on (B1

, g). The starting point of our analysis is the following result, due to Bethuel,

Orlandi and Smets in the case where g is the Euclidean metric.

Proposition 2.24 (Bethuel-Orlandi-Smets, [BOS05b]). There exists constants �0

2(0, 1/2) and C,↵

0

, ✏1

> 0, depending only on n,� and ⇤, such that if g 2 M1,�,⇤

and

u : (B1

, g) ! D is a solution to (2.1.2) with ✏ < ✏1

satisfying

(2.4.3)

ˆB1

e✏

(u) dvol ✏�↵0 ,

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 57

and that

(2.4.4) |u| � 1� �0

, on B1

.

Then, for x 2 B1/2

, we have

(2.4.5) e✏

(u)(x) C

ˆB1

e✏

(u) dvol .

In fact, a large part the proof of Proposition 2.24 given in [BOS05b] relied only on

elementary real analysis and the Calderon-Zygmund estimates for elliptic equations,

and can be adapted to the case of non-flat metrics in M1,�,⇤

in a straightforward

manner. Therefore, instead of repeating the whole proof, we will merely indicate the

modifications and refer the reader to Theorem A.2 of [BOS05b] for the rest of the

argument. Actually, the only point we believe requires some care is that the result of

Chen and Struwe ([CS89]) mentioned in remark A.3 of [BOS05b] should be replaced

by the following version.

Lemma 2.25. Let u : (B1

, g) ! C solve (2.1.2) with g 2 M1,�,⇤

. Then the following

hold.

(1) Then there exists a constant depending only on n such that there holds

(2.4.6) �e✏

(u) � |r2u|2 + Ric(ru,ru) + ✏�2

�

�r|u|2��2 � Ce✏

(u)2.

In particular,

(2.4.7) �e✏

(u) � �C (1 + e✏

(u)) e✏

(u),

where C depends on n and ⇤.

(2) There exists �0

and C, depending only on n,�,⇤, such that if´B1

e✏

(u) dvol <

�0

, then we have

(2.4.8) e✏

(u)(x) C

ˆB1

e✏

(u) dvol, for x 2 B1/2

.

Sketch of proof. The inequality (2.4.6) (in fact, a parabolic version) was first proved

by Chen and Struwe ([CS89]) when the metric is flat. A derivation can also be found

in [JS99], which we follow here. For the gradient term in e✏

(u), the standard Bochner

formula gives

1

2�|ru|2 = |r2u|2 + Ric(ru,ru) + hru,r�ui

= |r2u|2 + Ric(ru,ru)� 1� |u|2✏2

|ru|2 + |r|u|2|22✏2

,

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 58

where we used (2.1.2) in the second step. On the other hand, the computation for

the potential part of e✏

(u) is exactly the same as in the flat case, and we have

�

✓

1� |u|24✏2

◆

=|r|u|2|2

2✏2+

|u|2 � 1

✏2

✓

|ru|2 + |u|2 � 1

✏2|u|2◆

.

Adding up, we arrive at

�e✏

(u) = |r2u|2 + Ric(ru,ru) +|r|u|2|2✏2

+(1� |u|2)2

✏4|u|2 � 2

1� |u|2✏2

|ru|2.The proof of (1) is complete upon noticing that, using the Schwartz inequality,

(1� |u|2)2✏4

|u|2 � 21� |u|2✏2

|ru|2 � �4|ru|4 + (1� |u|2)2✏4

✓

|u|2 � 1

4

◆

�(

�4|ru|4 � �16e✏

(u)2, if |u|2 � 1

4

,

�4|ru|4 � C (1�|u|2)4✏

4 � �Ce✏

(u)2, if |u|2 < 1

4

.

For (2), we remark that this type of results are very common in geometric analysis,

and that there is a standard procedure for proving them using a Bochner-type identity

and a monotonicity formula. We refer the reader to [Sch84] for this argument, and

note that the inequalities (2.1) and (2.5) there should be replaced by our (2.4.7) and

(2.2.20), respectively. ⇤

Combining Proposition 2.24 and Theorem 2.17, we obtain

Corollary 2.26. There exists ⌘1

, ✏2

and C, depending only on n,� and ⇤, such that

if g 2 M1,�,⇤

and u : (B, g) ! D is a solution to (2.1.2) with

(2.4.9)

ˆB1

e✏

(u) dvol ⌘ |log ✏| ,

where ⌘ < ⌘1

and ✏ < ✏2

, then we have

(2.4.10) supx2B1/4

e✏

(u) C

ˆB1

e✏

(u) dvol .

Proof. For all x0

2 B3/4

, we have

(2.4.11) 4n�2

ˆB1/4(x0)

e✏

(u) dvol 4n�2

ˆB1

e✏

(u) dvol 4n�2⌘ |log ✏| .

Rescaling u, g and ✏ by letting

(2.4.12) u(x) = u(x0

+ x/4); gij

(x) = gij

(x0

+ x/4); ✏ = ✏/(1/4) = 4✏,

we see that g 2 M1,�,⇤

and u solves (2.1.2) with ✏ in place of ✏. More importantly,

by Theorem 2.17, if 4✏ < ✏0

and 4n�2⌘ < ⌘0

, then (2.3.16) holds and we get, upon

scaling back,

|u(x0

)| = |u(0)| � 1� C⌘s.

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 59

Now we set ⌘1

= min{42�n⌘0

, (�0

/C)1/s} (⌘0

, C as in Theorem 2.17, �0

as in Propo-

sition 2.24) and choose ✏2

< min{✏0

/4, ✏1

} (✏1

as in Proposition 2.24) so small that

|log ✏| ✓

3

4

◆

n�2

✓

4✏

3

◆�↵0

,

for all ✏ < ✏2

. Then we infer by the above arguments that |u| � 1 � �0

on B3/4

.

Moreover, there holds

(2.4.13)

✓

3

4

◆

2�n

ˆB3/4

e✏

(u) dvol ✓

4✏

3

◆�↵0

,

Thus, we get the desired conclusion upon rescaling u, g and ✏ as in (2.4.12) with 3/4

in place of 1/4, applying Proposition 2.24, and scaling back. ⇤

A final ingredient we need is the following proposition, which is a consequence

of the equation (2.1.2) in polar coordinates and the maximum principle. We refer

to Proposition A.1 of [BOS05b] and Lemma A.6 in [BOS05a] for a proof, which

applies to non-flat metrics as well.

Proposition 2.27 ([BOS05b], Proposition A.1). Let g 2 M1,�,⇤

and assume that

u : (B1

, g) ! D is a solution to (2.1.2), with |u| � 1/2 on B1

. Then, in fact

(2.4.14) |u(x)| � 1� C✏2�

1 + |ru|20;B1

�

, for all x 2 B1/2

, and for ✏ < ✏3

where C and ✏3

depend only on n,� and ⇤.

We now commence the proof of statements (1) and (2) of Theorem 1.7. Again,

the proof is quite long and will be divided into a number of steps.

Proof of Theorem 1.7(1)(2). By the compactness of M and the smoothness of g we

see that there exists r0

, � and ⇤, depending only on M and its metric g, such that on

each geodesic ball Br0(x) ⇢ M , the components (g

ij

) of g with respect to the normal

coordinates satisfy

(1) gij

(0) = �ij

and [gij

]1,1;B

r0 ⇤ for all i, j;

(2) �|⇠|2 gij

(x)⇠i⇠j ��1|⇠|2, for all x 2 Br0 , ⇠ 2 Rn,

where we’ve identified Br0(x) ⇢ M with B

r0 ⇢ Rn via the exponential map. Without

loss of generality, we will assume below that r0

< 1.

To set up the proof, we first note that by (1.2.6), {µ✏

} is a bounded sequence of

Radon measures, and thus there exists a subsequence {µ✏

k

} converging weakly to a

limit measure, which we denote by µ. We now fix a positive number ⌘ depending

only on n,� and ⇤ by

⌘ =1

2min{⌘

1

, 42�n, (2C)�1/s}.

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 60

The constant ⌘1

is given by Corollary 2.26 with � and ⇤ as above, and C, s are given

by Theorem 2.17 with µ = 1 and �,⇤ as above.

The set ⌃ is then defined as

(2.4.15) ⌃ = {x 2 M | lim infr!0

r2�nµ(Br

(x)) � ⌘/3}.Below, we first show that ⌃ has the asserted properties except for the (n � 2)-

rectifiability. Then we derive estimates on {u✏

} away from ⌃ and identify the 1-form

in the statement of Theorem 1.7(1). Finally we establish the decomposition (2.4.1),

prove (2.4.2) and verify that ⌃ is (n� 2)-rectifiable.

Step 1: Properties of µ and ⌃.

We now show that Hn�2(⌃) < 1. By the definition, for all x 2 ⌃ there exists

rx

2 (0, r0

) such that

(2.4.16) r2�nµ(Br

(x)) > ⌘/4, for all r < rx

.

This implies that the collection

(2.4.17) B ⌘ {Br

(x)| x 2 ⌃, r2�nµ(Br

(x)) � ⌘/4}covers the set ⌃ finely, in the sense defined in p.12 of [Sim83]. It now follows from

a standard covering argument that Hn�2(⌃) (cn

/⌘)µ(M) < 1.

The remainder of Step 1 is devoted to establishing two further properties of the

limit measure µ: One is the following monotonicity result, the other appears as

Lemma 2.30 and gives density upper- and lower-bounds.

Lemma 2.28. The function

r 7! e�0rr2�nµ(B

r

(x))

is non-decreasing for r 2 (0, r0

) for all x 2 M . Here �0 = �/r0

and � = �(n,�,⇤) is

given by Proposition 2.10 with � and ⇤ as in the beginning of the proof of Theorem

1.7.

Proof. Fix x0

2 M , we introduce normal coordinates on Br0(x) and identify it with

the Euclidean ball Br0 . Next we rescale g, u

✏

k

, ✏k

, µ✏

k

and µ as follows in order to

reduce to the situation considered in Proposition 2.10:

uk

(x) = u✏

k

(r0

x), (gk

)ij

(x) = gij

(r0

x), ✏k

= ✏k

/r0

,

µk

(A) = r2�n

0

µ✏

k

(r0

A), µ(A) = r2�n

0

µ(r0

A),

where A is any Borel set in B1

. It is then easy to see that, for each r < 1,

(2.4.18) r2�nµ(Br

) = (rr0

)2�nµ(Brr0(x)).

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 61

Moreover, observe that under the above rescaling, each uk

: (B1

, gk

) ! D solves

(2.1.2) with ✏ = ✏k

, and each gk

lies in M1,�,⇤

. Therefore Proposition 2.10 applies.

In particular, (2.2.1) shows that, for each k, the following function is non-decreasing.

(2.4.19) r 7! e�rr2�nµk

(Br

).

To see that the above monotonicity holds with µ in place of µk

, observe that since

µ is the weak limit of {µk

}, for all but countably many r there holds

(2.4.20) limk!1

µk

(Br

) = µ(Br

).

Hence, taking a pair of radii 0 < � < ⇢ < r0

, we may choose two sequences {�i

} and

{⇢i

}, tending to � and ⇢, respectively, such that (2.4.20) holds with r = �i

and r = ⇢i

for all i. Now we observe that, for each fixed i (we may assume that �i

< ⇢i

for all i),

e�⇢⇢2�nµ(B⇢

) � e�(⇢�⇢i)✓

⇢i

⇢

◆

n�2

e�⇢i⇢2�n

i

µ(B⇢

i

)

= e�(⇢�⇢i)✓

⇢i

⇢

◆

n�2

limk!1

e�⇢i⇢2�n

i

µk

(B⇢

i

)

� e�(⇢�⇢i)✓

⇢i

⇢

◆

n�2

limk!1

e��i�2�n

i

µk

(B�

i

)

= e�(⇢�⇢i)✓

⇢i

⇢

◆

n�2

e��i�2�n

i

µ(B�

i

)

� e�(⇢�⇢i)✓

⇢i

⇢

◆

n�2

e�(�i��)✓

�

�i

◆

n�2

e���2�nµ(B�

).

Letting i tend to infinity verifies that the function (2.4.19) with µ in place of µk

is

non-decreasing for r 2 (0, r0

). Scaling back and recalling (2.4.18), we get the desired

result. ⇤

Remark 2.29. Of course, the proof of Lemma 2.28 also shows that, for all k,

(2.4.21) r ! e�0rµ

✏

k

(Br

(x))

is non-decreasing for r 2 (0, r0

) for all x 2 M . One has only to use the monotonicity

of (2.4.19) and scale back to see this.

To state the next Lemma we introduce the density function of µ:

(2.4.22) ⇥(µ, x) ⌘ limr!0

r2�nµ(Br

(x)).

Note that Lemma 2.28 shows that the limit above always exists.

Lemma 2.30. For µ-a.e. x 2 M , the limit in (2.4.22) exists and is non-zero and

finite. In other words, ⇥(µ, x) 2 (0,1) for µ-a.e. x 2 M .

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 62

Proof of Lemma 2.30. Take a point x 2 ⌃. Notice that for each ⇢ < r0

/4, we can

choose s 2 (⇢, 2⇢) such that (2.4.20) holds with r = s. Then we easily see with the

help of Lemma 2.28 that

⇢2�nµ(B⇢

) e�0(s�⇢)s2�nµ(B

s

) = e�0(s�⇢) lim

k!1s2�nµ

✏

k

(Bs

)

e�0(r0�⇢) lim

k!1r2�n

0

µ✏

k

(Br0)

e�0r0r2�n

0

lim supk!1

1

|log ✏k

|ˆM

e✏

(u✏

k

) dvol .

The quantity in the last line is finite by the assumption (1.2.6). Thus, letting ⇢

descend to zero, we infer that ⇥(µ, x) < 1. Finally, the fact that ⇥(µ, x) > 0,

follows from the definition of the set ⌃. Thus, Lemma 2.30 is proved. ⇤

For later purposes, we note that in the proof of Lemma 2.30, we actually obtained

(2.4.23) ⇥(µ, x) 2 (⌘/4, E0

), for x 2 ⌃,

where E0

⌘ e�0r0r2�n

0

lim supk!1

1

|log ✏k

|´M

e✏

(u✏

k

) < 1.

Step 2: Analysis away from ⌃.

Fix a point x0

/2 ⌃, then by the definition of ⌃ and Lemma 2.28, there exists a

threshold kx0 and a radius r = r

x0 which we can assume to be less than r0

, such that

for k � kx0 and s < r, there holds

(2.4.24) s2�nµ✏

k

(Bs

(x0

)) < ⌘/3.

We identify the geodesic ball Br

(x0

) with Br

⇢ Rn via the exponential map and

perform the following rescaling:

(2.4.25) uk

(x) = u✏

k

(rx); (gk

)ij

(x) = gij

(rx); ✏k

= ✏k

/r.

Then we again reduce to the local situation considered above and in the previous

two sections. Below, unless otherwise stated, all estimates are meant to hold for

su�ciently large k.

Recalling the definition of µ✏

k

, we see that under the above rescaling, (2.4.24)

becomes

lim infk!1

1

|log ✏k

|ˆB1

|ruk

|22

+(1� |u

k

|2)4✏2

k

dvolg

k

< ⌘/3.

Thus, since r = rx0 is a fixed number, for su�ciently large k we have

(2.4.26)

ˆB1

|ruk

|22

+(1� |u

k

|2)4✏2

k

dvolg

k

<⌘

3|log ✏

k

| < ⌘ |log ✏k

| .

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 63

Since we are requiring ⌘ < ⌘1

, the gradient estimate (2.4.10) holds, and hence

(2.4.27) |ruk

(x)|2 C

ˆB1

e✏

k

(uk

) dvolg

k

< C |log ✏k

| , for x 2 B1/4

.

Furthermore, since ⌘ < min{42�n, (2C)�1/s}, we deduce as in the first part of the

proof of Corollary 2.26 that |uk

(x)| � 1/2, for x 2 B3/4

, and thus by Proposition

2.27, we have

(2.4.28)1� |u

k

(x)|✏2k

C⇣

1 + |ru|20;B3/4

⌘

C |log ✏k

| , for x 2 B1/4

.

To proceed, note that since uk

doesn’t vanish on B1/4

, we may introduce polar

coordinates on C and write uk

= ⇢k

ei'k . Then on B1/4

the equation 2.1.2 splits into

��⇢k

+ ⇢k

|r'k

|2 = ✏�2

k

(1� ⇢2k

)⇢k

,(2.4.29)

�'k

+ 2⇢�1

k

hr⇢k

,r'k

i = 0.(2.4.30)

Note that we can without loss of generality assume that�B1/4

'k

dvolg

k

2 [0, 2⇡). Thus

'k

must take value in [0, 2⇡) somewhere in B1/4

. From this fact and the estimate

(2.4.27), we infer that

(2.4.31) |'k

(x)| C |log ✏k

|1/2 , for x 2 B1/4

.

On the other hand, letting ✓k

= 1 � ⇢k

, we infer from (2.4.28) and (2.4.29) that

|✓k

|0;B1/4

C ✏2k

|log ✏k

|, and that |�✓k

|0;B1/4

C |log ✏k

|. Hence, by (2.4.69) in Lemma

2.35 below, we have

(2.4.32) |r⇢k

|0;B1/5

= |r✓k

|0;B1/5

C ✏k

|log ✏k

| .We will bootstrap the above basic estimates in the following Proposition.

Proposition 2.31. Letting Xk

= ✏�2

k

(1� ⇢k

). There holds, for l = 0, 1, 2, · · · , and k

large enough,

|r'k

|l,0;B2�2l�2

Cl

|log ✏k

|1/2 ,(2.4.33)

|Xk

|l,0;B2�2l�2

Cl

|log ✏k

| .(2.4.34)

|r⇢k

|l,0;B2�2l�3

Cl

✏↵l

k

|log ✏k

|�l , ,(2.4.35)

where ↵l

and �l

are positive constants depending only on l.

Remark 2.32. The structure of the proof of 2.31 follows [BBH93]. However, there

the authors were concerned with proving bounds independent of ✏, whereas we have

to keep track of the |log ✏|-factors.Proof of Proposition 2.31. The proof will proceed by induction. The case l = 0 has

already been taken care of above. Therefore we suppose (2.4.33), (2.4.34) and (2.4.35)

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 64

hold for 0, · · · , l and prove them for l + 1. For that purpose, we fix radii 2�2l�4 <

r4

< r3

< r2

< r1

< 2�2l�3, since the inductive step consists of a number of interior

estimates. Throughout this proof, Cl

, ↵l

and �l

will denote positive constants that

depend only on l, and may change from line to line.

To begin, we see from the induction hypotheses that

|�'k

|l;B2�2l�3

C ✏↵l

k

|log ✏k

|�l+1/2 ,(2.4.36)

|�✓k

|l;B2�2l�2

= |�⇢k

|l;B2�2l

C |log ✏k

| .(2.4.37)

By (2.4.36) and the higher-order interior Schauder estimates,

(2.4.38) |'k

|l+1,1/2;B

r1 C

⇣

|'k

|0;B1/4

+ |�'k

|l;B2�2l�3

⌘

C |log ✏k

|1/2 .On the other hand, by (2.4.28), (2.4.37) and the multiplicative Schauder estimates

we get

(2.4.39) |r⇢k

|l,1/2;B2�2l�3

|✓k

|l+1,1/2;B2�2l�3

C ✏↵l

k

|log ✏k

|�l .Using (2.4.38) and (2.4.39) in (2.4.30) yields

(2.4.40) |�'k

|l,1/2;B

r1 C

l

✏↵l

k

|log ✏k

|�l .Thus, by the interior Schauder estimates again,

(2.4.41) |'k

|l+2,1/2;B

r2 C

l

⇣

|'k

|0;B1/4

+ |�'k

|l,1/2;B

r1

⌘

Cl

|log ✏k

|1/2 .In particular, this proves (2.4.33) with l + 1 in place of l.

Next we turn to proving (2.4.34) for l + 1. To that end, notice that Xk

satisfies

the equation

(2.4.42) ✏2k

�Xk

= �⇢k

|r'k

|2 +Xk

(1 + ⇢k

)⇢k

.

Thus, by induction hypotheses,

(2.4.43) |�Xk

|l;B2�2l�2

Cl

✏�2

k

|log ✏k

| .Hence the interior Schauder estimates yield

(2.4.44) |Xk

|l+1;B2�2l�3

Cl

✏�2

k

|log ✏k

| .Now we take a multi-index ↵ with |↵| = l, apply @↵ to both sides of (2.4.42) and use

(2.4.74), obtaining

|�@↵Xk

|0;B2�2l�3

|@↵�Xk

|0;B2�2l�3

+ Cl

|Xk

|l+1;B2�2l�3

(2.4.45)

Cl

✏�2

k

|log ✏k

| ,

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 65

where we used (2.4.43) and (2.4.44) in the second inequality. Therefore, we infer from

Lemma 2.35 and (2.4.34) that

|r@↵Xk

|0;B

r1 C

⇣

|@↵Xk

|0;B2�2l�3

+ |@↵Xk

|1/20;B2�2l�3

|�@↵Xk

|1/20;B

2�2l�3

⌘

Cl

✏�1

k

|log ✏k

| .Since ↵ is any multi-index with |↵| = l, we obtain

(2.4.46) |Xk

|l+1;B

r1 C

l

✏�1

k

|log ✏k

| .Next we improve (2.4.46) to get (2.4.34) for l + 1. Before that, let us observe that

we can use (2.4.46) to get a C l+2-estimate on Xk

, which will be useful for handling

commutator terms in the subsequence argument. Specifically, looking back at (2.4.42),

we see from (2.4.35), (2.4.41) and (2.4.46) that

(2.4.47) |�Xk

|l+1;B

r2 C

l

✏�3

k

|log ✏k

| .Therefore, interior Schauder estimates give

(2.4.48) |Xk

|l+2;B

r3 C

l

✏�3

k

|log ✏k

| .Applying any partial derivative @↵ with |↵| = l + 1 to (2.4.42) and using (2.4.47),

(2.4.48) along with (2.4.74), we arrive at the following analogue of (2.4.45).

(2.4.49) |�@↵X|0;B

r3 C

l

✏�3

k

|log ✏k

| .From (2.4.49), (2.4.46) and Lemma 2.35 we infer as above that

(2.4.50) |Xk

|l+2;B

r4 C

l

✏�2

k

|log ✏k

| .Now we proceed to improve (2.4.46) to the desired estimate. To that end, we

rewrite (2.4.42) as

(2.4.51) � ✏2k

�Xk

+ 2Xk

= ⇢k

|r'k

|2 + 3✏2k

X2

k

� ✏4k

X3

k

⌘ Rk

.

Note that by (2.4.34), (2.4.41) and (2.4.46), we have |Rk

|l+1;B

r2 C

l

|log ✏k

|. Now, wetake a multi-index ↵ of length l+1 and apply it to both sides of the above equation,

getting

(2.4.52) � ✏2k

�@↵Xk

+ 2@↵Xk

= ✏2k

[@↵,�]Xk

+ @↵Rk

.

With the help of (2.4.74) and the estimates obtained so far, we see that we are in the

situation of Lemma 2.36, with v = @↵X, R = r4

and

A =|@↵Xk

|0;B

r4 |@↵X

k

|0;B

r2 C

l

✏�1

k

|log ✏k

|F =

�

�✏2k

[@↵,�]Xk

+ @↵Rk

�

�

0;B

r4 C

l

✏2k

|Xk

|l+2;B

r4+ |R

k

|l+1;B

r2

Cl

|log ✏k

| .

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 66

Note that we used (2.4.50) and the fact that |Rk

|l+1;B

r2 C

l

|log ✏k

| in the last

inequality. We now invoke Lemma 2.36 to arrive at

(2.4.53) |@↵X|0;B2�2l�4

F/2 + Aep�

2�4l�8�r

24

2✏r4 Cl

|log ✏k

| .

Note that in the last inequality, we used the fact that the term Aep�

2�4l�8�r

24

2✏r4 is negli-

gible compared to |log ✏k

| since r4

is strictly larger than 2�2l�4 and is fixed depending

only on l. Since ↵ is any multi-index with length l + 1, we conclude that (2.4.34)

holds for l + 1.

Finally we prove (2.4.35) for l + 1. Recall that we denoted 1� ⇢k

by ✓k

and that

|✓k

|0;B1/4

C ✏2k

|log ✏k

|. From (2.4.29), (2.4.41) and (2.4.53), we infer that

(2.4.54) |�✓k

|l+1;B2�2l�4

= |�⇢k

|l+1;B2�2l�4

Cl

|log ✏k

| .Thus by Lemma 2.35 we get

(2.4.55) |✓k

|l+2;B2�2l�5

Cl+1

✏↵

l+1

k

|log ✏k

|�l+1 ,

and the estimate (2.4.35), with l + 1 in place l, follows immediately. By induction,

we have completed the proof of the Proposition. ⇤

Step 3: Identifying the harmonic 1-form

Thanks to the estimates in Proposition in 2.31, we can now identify the 1-form

in the statement of Theorem 1.7. To that end we first notice that, since uk

= ⇢k

ei'k ,

we have duk

= ei'k (d⇢k

+ i⇢k

d'k

). A direct computation then shows that

uk

⇥ duk

⌘ u1

k

du2

k

� u2

k

du1

k

= �Im�

uk

· duk

�

= ⇢2k

d'k

.(2.4.56)

If we now write

⇢2k

d'k

= d'k

� (1� ⇢2k

)d'k

= d'k

� ✏2k

Xk

d'k

,

then we see from Proposition 2.31 and (2.4.56) that, passing to a subsequence for

each l and taking a diagonal subsequence if necessary, there exists a 1-form so that

(2.4.57)uk

⇥ duk

|log ✏k

|1/2! in C l(B

2

�2l�2), l = 0, 1, 2, · · ·

Claim. is a smooth harmonic 1-form in B1/16

which is d-closed and d⇤-closed.

Proof of the claim. Notice that by (2.1.2) and (2.4.35), we have

d⇤(uk

⇥ duk

) = 0.(2.4.58)

d

uk

⇥ duk

|log ✏k

|1/2!

=2⇢

k

d⇢k

^ d'k

|log ✏k

|1/2! 0 in C0(B

2

�3).(2.4.59)

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 67

Therefore we see that, for all smooth 1-forms ⇣ compactly supported in B1/16

, we

have

(2.4.60) D( , ⇣) = limk!1

D

uk

⇥ duk

|log ✏k

|1/2, ⇣

!

= 0.

Hence is weakly harmonic in B1/16

, and the regularity theory of elliptic systems

implies that is in fact smooth in B1/16

. Finally, (2.4.57), (2.4.58) and (2.4.59) show

that d⇤ = 0 and d = 0 on B1/16

. The claim is proved. ⇤

Next we see how the energy of relates to the Ginzburg-Landau energy of uk

.

For that purpose, we recall (2.3.30) and the definition of Xk

to write

|ruk

|22

+(1� |u

k

|2)24✏2

k

=|u

k

⇥ duk

|22

+|r|u

k

|2|28

+(1� |u

k

|2)|ruk

|22

+(1� |u

k

|2)24✏2

k

=|u

k

⇥ duk

|22

+⇢2k

|r⇢k

|22

+ ✏2k

Xk

|ruk

|22

+ ✏2k

X2

k

4.

Dividing through by |log ✏k

| and using (2.4.57) along with Proposition 2.31, we infer

that, in the sense of measures on B1/16

(2.4.61) limk!1

1

|log ✏k

|✓ |ru

k

|22

+(1� |u

k

|2)24✏2

k

◆

dvolg

k

=| |22

dvolg

k

.

Before returning to the global picture, let us undo the rescaling (2.4.25) and

briefly summarize in the following proposition what we have achieved by the above

local analysis.

Proposition 2.33. Given x0

/2 ⌃, there exist a radius r = rx0 such that any subse-

quence {✏0k

} of {✏k

} verifies, for large enough k, the inequality (2.4.24) for all s < r.

Moreover, out of each such {✏0k

}, we can extract a further subsequence, still denoted

{✏0k

}, such that

(1) (cf. (2.4.57)) For each l � 0,

u✏

0k

⇥ du✏

0k

|log ✏0k

| converges in C l to some 1-form x0 on B

2

�2l�2r

(x0

).

(2) (cf. Claim after (2.4.57)) The form x0 is d-closed, d⇤-closed and harmonic in

Br/16

(x0

).

(3) (cf. (2.4.61)) In the sense of Radon measures on Br/16

(x0

),

limk!1

µ✏

0k

= limk!1

1

|log ✏0k

|

|ru✏

0k

|22

+(1� |u

✏

0k

|2)24✏02

k

!

dvolg

=|

x0 |22

dvolg

,

where the first equality is by definition.

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 68

Remark 2.34. To be completely precise, what we’ve done in the local setting actually

gives statements (1) and (3) above with |log(✏0k

/r)| in the denominator instead of

|log ✏0k

|. However, since r = rx0 is a fix radius, we have

(2.4.62) limk!1

|log(✏0k

/r)||log ✏0

k

| = limk!1

|log ✏0k

|� |log r||log ✏0

k

| = 1.

Thus it doesn’t matter whether we use |log ✏0k

| or |log(✏0k

/r)| in these statements.

Using Proposition 2.33 along with partitions of unity, and passing to further sub-

sequences of {✏k

} (which we still denote by {✏k

}) if necessary, we see that there existsa 1-form , defined on M \⌃, such that (1), (2) and (3) of Proposition 2.33 hold with

any compact subset K of M \ ⌃ in place of the balls B2

�2l�2r

(x0

) and Br/16

(x0

), and

with and ✏k

in place of x0 and ✏0

k

, respectively. In particular, we see that

(2.4.63) limk!1

µ✏

k

x(M \ ⌃) = | |22

dvolg

,

and thus for all compact subset K ⇢⇢ M \ ⌃, there holds

(2.4.64)

ˆK

| |22

dvolg

lim supk!1

µ✏

k

(K) < 1.

Hence we infer that is L2 on M .

We next show by a capacity argument that is in fact harmonic across ⌃. Some

preparations are in order: Let {Wk

} be a nested sequence of open sets containing ⌃,

with W1

� W2

� · · ·⌃. Since Hn�2(⌃) < 1, as in the proof of Theorem 3, Section

4.7.2 of [EG92], we obtain a sequence of cut-o↵ functions {⇣k

} such that

(1) supp ⇣k

⇢ Wk

.

(2) ⇣k

⌘ 1 on Wk+1

.

(3) Most importantly, we can arrange so thatˆW

k

|r⇣k

|2 dvol < 1/k.

Now, to see that is harmonic on M , we first show that is weakly d-closed on

all of M . For that purpose we take an arbitrary smooth 2-form ⇠ on M and compute

0 =

ˆM

h , d⇤ ((1� ⇣k

)⇠)i dvol

=

ˆM

(1� ⇣k

)h , d⇤⇠i dvol�ˆB

h , ⇠xd⇣k

i dvol

⌘ I � II,

where the equality in the first line follows from property (2) of {⇣k

} and the fact that

d = 0 away from ⌃. Letting k tend to infinity, from property (1) above and the fact

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 69

that 2 L2(M), we infer that

limk!1

(I) =

ˆM

h , d⇤⇠i dvol .

On the other hand, from property (3) and the Holder inequality, we estimate

|(II)| Ck k2;M

kr⇣k

k2;W

k

Ck�1k k2;M

! 0 as k ! 1.

Therefore, we conclude that

(2.4.65)

ˆM

h , d⇤⇠i dvol = 0, for all smooth 2-form ⇠.

In other words, is weakly d-closed.

To proceed, since is L2 by (2.4.64), we may consider its Hodge decomposition

on M :

(2.4.66) = d↵ + h,

where ↵ 2 W 1,2(M ;R) and h is a harmonic 1-form on M . Note that there is no

d⇤-term since is weakly d-closed on M .

Claim. d↵ = 0

Proof of the claim. Take a smooth test function v 2 C1

c

(M \ ⌃) and computeˆM

hd↵, dvi dvol =ˆM\⌃

h , dvi �ˆM

hh, dvi dvol = 0.

The first term on the right-hand side vanishes because d⇤ = 0 on M \ ⌃, whereas

the second term vanishes because any 1-form harmonic on all of M is automatically

d- and d⇤-closed. Therefore ↵ is weakly harmonic on M \ ⌃. Cutting o↵ using the

test functions {⇣k

} introduced above and following the proof of the weak d-closedness

of on M , we conclude that ↵ is indeed weakly harmonic on all of M , but then ↵

must be a constant since M is closed, and hence d↵ = 0. The claim is proved. ⇤

Returning to (2.4.66), by the Claim we see that = h, and thus is harmonic

on all of M . In particular, by elliptic regularity, is smooth.

Step 4: Decomposition of µ and rectifiability of ⌃. By (2.4.63), we see that if

we let

(2.4.67) ⌫ = µ� | |22

dvolg

,

then supp ⌫ ⇢ ⌃. Thus it remains to prove that limr!0

r2�n⌫(Br

(x0

)) exists for Hn�2-a.e.

x 2 ⌃, and that (2.4.2) holds. In view of (2.4.23), this boils down to proving that,

for Hn�2-a.e. x 2 ⌃, there holds

(2.4.68) limr!0

r2�n

ˆB

r

(x)

| |2 dvol = 0.

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 70

However, since is a smooth harmonic form on all of M , the condition (2.4.68)

is obviously true everywhere on M . Thus (2.4.2) holds as well. Thanks to the

density bounds (2.4.2) and the fact that ⌫ is supported on ⌃, we can now invoke the

rectifiability theorem of Preiss [Pre87] to conclude that ⌃ is rectifiable, and we have

at last completed the proof of Theorem 1.7(1)(2).

⇤

Here we gather some technical tools that we used in the proof of Theorem 1.7(1)(2).

Note that all three of the Lemmas below are either standard or follow easily from

standard material. However, it is perhaps not easy to find them in the literature.

Therefore we will briefly indicate how each result is proved. The first result is a

multiplicative version of the interior Schauder estimates for elliptic equations.

Lemma 2.35. There exist a constant C, depending only on n, l, |gij

|l,1/2;B1 and s

such that for u 2 C l+1,1/2(B1

;C), there hold

|Dl+1u|0;B

s

C(|u|0;B1 + |u|

1l+2

0;B1|Dl�

g

u|l+1l+2

0;B1),(2.4.69)

[Dl+1u]0,1/2;B

s

C(|u|0;B1 + |u|

12l+4

0;B1|Dl�

g

u|2l+32l+4

0;B1).

We remark that the proof given in [BBH93] for the first estimate with l = 0

can be followed to obtain both estimates for all l. The main idea is to derive the

multiplicative version from the additive version using a scaling argument. We will

omit the details.

The next result we need is a special type of maximum principle, originally due to

[BBH93] in the case of two-dimensional domains in R2. We extend their argument

to the following more general setting.

Lemma 2.36 (cf. [BBH93], Lemma 2). Let g 2 M�,⇤

and A,F be two positive

constants. Suppose v : B1

⇢ Rn ! C is a solution to(

�✏2�g

v + 2v = f , on BR

v = g , on @BR

,

where |f |0;B

R

F and |g|0;@B

R

A. Then we have

(2.4.70) |v|0;B

R

F

2+ Ae

p�

|x|2�R

2

2✏R ,

provided ✏ <�

n��1/2/R + C�1/2��1

, where C depends only on n,�,⇤.

Proof. Straightforward computation shows that

�g

✓

ep�

|x|2�R

2

2✏R

◆

=⇣

gij@ij

+p

det(g)�1

@i

(p

det(g)gij)@j

⌘

✓

ep�

|x|2�R

2

2✏R

◆

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 71

= ep�

|x|2�R

2

2✏R

gii

p�

✏R+�gijx

i

xj

✏2R2

+ @i

(p

det(g)gij)

p�x

j

✏R

!

ep�

|x|2�R

2

2✏R

n��1/2

✏R+

1

✏2R2

+ Cn,�,⇤

p�

✏

!

,

where we have estimated |x| by R in the last two terms in the last inequality. Thus,

letting w denote the right-hand side of (2.4.70), we infer that

(2.4.71) � ✏2�g

w+ 2w � F +Aep�

|x|2�R

2

2✏R

1� ✏

✓

n��1/2

R+ C

n,�,⇤

�1/2◆�

� F � f,

provided ✏ <�

n��1/2/R + C�1/2��1

. On the other hand, clearly we have w = F/2 +

A � A � g on @BR

. The standard maximum principle now yields the result. ⇤

The last technical device is essentially a standard computation that gives estimates

on the commutator of �g

with any partial derivative. We state the lemma in multiple

parts just for clarity, and in practice we only use assertion (3) below.

Lemma 2.37. (1) Let T be a smooth p-covariant tensor on (B1

, g). Then

(rk

�T )i1,··· ,ip =(�rT )

k,i1,··· ,ip �Rl

k

(rT )l,i1,··· ,ip(2.4.72)

�p

X

⌫=1

gijRl

kii

⌫

(rT )j,··· ,i

⌫�1,l,i⌫+1,··· ,ip ,

where Rl

k

and Rl

kii

⌫

denote the Ricci and Riemannian curvature tensors, respec-

tively.

(2) Let T be as above, then we have the following pointwise estimate.

(2.4.73)�

�[�,rk]T�

� CX

1jk

|rjT |,

where C depends only on n, k and |gij

|k+1;B1.

(3) Let u 2 Ck+2

loc

(B1

;C), and let @↵ with |↵| = k denote a k-th order partial

derivative in coordinates. Then we have

(2.4.74) |[�, @↵]u| CX

1jk+1

|rju|,

where C depends only on n, k and |gij

|k+1;B1.

Proof. For (1), we compute by definition.

(rk

�T )i1,··· ,ip = gijr3

k,i,j

Ti1,··· ,ip

= gij

r3

i,k,j

Ti1,··· ,ip �Rl

kij

rl

Ti1,··· ,ip �

p

X

⌫=1

Rl

kii

⌫

rj

T··· ,i⌫�1,l,i⌫+1,···

!

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 72

= gijr3

i,j,k

Ti1,··· ,ip �Rl

k

rl

Ti1,··· ,ip �

p

X

⌫=1

gijRl

kii

⌫

rj

T··· ,i⌫�1,l,i⌫+1,···.

Thus we obtain the desired identity.

Now (2) follows from iterating (2.4.72). Specifically, we will argue by induction.

The case k = 1 follows immediately from (2.4.72). Next, suppose (2.4.73) holds for

k = m, then we notice that

[�,rm+1]T = [�,rm]rT +rm([�,r]T ).(2.4.75)

By the induction hypothesis, the first term on the right-hand side is bounded by

|[�,rm]rT | CX

1jm

|rj(rT )|,

whereas the second term is bounded by

|rm([�,r]Tk,i1,··· ,ip)| =

�

�

�

�

�

rm

Rl

k

rl

Ti1,··· ,ip +

p

X

⌫=1

gijRl

kii

⌫

rj

T··· ,i⌫�1,l,i⌫+1,···

!

�

�

�

�

�

CX

1jm+1

|rjT |.

Combining the two above inequalities and recalling (2.4.75) yields the result.

Finally, assuming that @↵ = @

@x

↵1· · · @

@x

↵

k

,(3) follows from (2.4.73) with u in place

of T and the following observations, which can be verified by straightforward compu-

tations and hence we omit the details.�

�(�rku)↵1,··· ,↵

k

��(@↵u)�

� CX

1jk+1

|rju|,�

�rk

↵1,··· ,↵k

(�u)� @↵�u�

� CX

1jk+1

|rju|.

Note that C again only depends on n, k and |gij

|k+1;B1 . ⇤

We now turn to conclusion (4) of Theorem 1.7. For that purpose we need the

help of Proposition 2.38 below. Recalling that H2

is the space of harmonic 2-forms

on M , we let W 1,2

2,?(M) and L2

2,?(M) denote the L2-orthogonal complements of H2

in

W 1,2

2

(M) and L2

2

(M), respectively. Also, W�1,2

2,? (M) will denote the set of bounded

linear functionals on W 1,2

2

(M) that annihilates H2

. When there no risk of confusion,

we often drop the manifold M from the notations.

Proposition 2.38. For all w 2 L2

2,? there exists a unique form ' 2 W 1,2

2,? satisfying

(2.4.76) D(', ⇣) =

ˆM

hw, ⇣i dvol, for all ⇠ 2 W 1,2

2

.

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 73

Moreover, for p 2 [1, n/(n� 1)) we have the following estimate.

(2.4.77) kd⇤'kp;M

Ckwk1;M

,

where C depends only on n, p and the metric g.

Proof. It is standard that, given w 2 L2

2,?, there exists a unique solution in W 1,2

2,? to

(2.4.76). Thus we will focus on getting the estimate (2.4.77). The idea will be to use

a duality argument.

Specifically, letting p0 = p/(p � 1) and recalling the converse of the Holder in-

equality, we have

kd⇤'kp;M

= sup

⇢ˆM

hd⇤', hi dvol | h 2 Lp

0

1

, khkp

0;M

= 1

�

.

Given an h as in the above definition, we consider the following auxiliary problem

(2.4.78) D(⇠, ⇣) =

ˆM

hh, d⇤⇣i dvol, for all ⇣ 2 W 1,2

2

.

Note that the right-hand side of the above equation lies in W�1,2

2,? and hence, again by

standard theory, there exists a unique solution ⇠ 2 W 1,2

2

. Moreover, by the Calderon-

Zygmund estimates for elliptic systems, we know that

(2.4.79) k⇠k1;M

Ck⇠k1,p

0;M

Ckhkp

0;M

C

where the first inequality follows by the Sobolev embedding since p0 > n, and the last

inequality holds since we are assuming that khkp

0;M

= 1. Now we test the equation

(2.4.76) against this ⇠, obtainingˆM

hw, ⇠i dvol = D(', ⇠) =

ˆM

hh, d⇤'i dvol,

where we used the equation (2.4.78) for the second equality. Therefore, we infer that�

�

�

�

ˆM

hh, d⇤'i dvol�

�

�

�

k⇠k1;M

kwk1;M

Ckwk1;M

.

Since h is an arbitrary 1-form with khkp

0;M

= 1, we conclude that kd⇤'kp;M

Ckwk

1;M

, as is to be shown. ⇤

Remark 2.39. The proof we just gave is adapted from that of Proposition A.2 in

[BBO01].

For the remainder of this section, we assume that M is simply-connected.

Equipped with Proposition 2.38, we can now following the arguments in Section VI

of [BBO01] to obtain the following global W 1,p-estimate for p 2 (1, n/(n� 1)) which

will play an essential role to the proof of Theorem 1.7(4).

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 74

Proposition 2.40. Let u : M ! C be a solution to (2.1.2), and suppose that there

holds

(2.4.80)1

|log ✏|ˆM

e✏

(u) dvolg

E1

.

Then for p 2 [1, n/(n� 1)), there exists a constant C = C(M, g, p, E1

) such that

(2.4.81)

ˆM

|ru|p dvol C.

Sketch of proof. The proof we give here follows closely the proof of estimate (7) of

(BBO). We will focus on the modifications we made and sketch the rest of the argu-

ment.

Recalling (2.3.30), we write

|u|2|ru|2 = 1

4

�

�r|u|2��2 + |u⇥ru|2.We will estimate the two terms on the right-hand side and then indicate how to

conclude the proof. The proof will resemble that of Theorem 2.18, only that now we

are working globally.

Step 1: Estimates for u⇥ du.

As in the proof of Theorem 2.18, the main tool is the Hodge decomposition.

Specifically, we let ↵ be the same cut-o↵ function as in Step 2 in the proof of Theorem

2.14 and perform a Hodge decomposition on the 1-form ↵u⇥du, obtaining a function

h 2 W 1,2(M ;R) and a 2-form ' 2 W 1,2

2

such that

(2.4.82) ↵u⇥ du = dh+ d⇤'.

Without loss of generality, we may assume that h integrates to zero on M , and that

' 2 W 1,2

2,?. Applying d to both sides of (2.4.82), we get

�' = d(↵u⇥ du) ⌘ w.

By (2.3.39), Proposition 2.22 and the upper bound (2.4.80), we know that

kwk1;M

C

ˆ{|u|1��}

(1� |u|2)2✏2

C.

Also, since w is exact, we infer that it is in L2

2,?, and hence Proposition 2.38 gives

(2.4.83) kd⇤'kp;M

Ckwk1;M

C.

This takes care of ', and we next turn to h, which is easily seen to satisfy

d⇤(↵�1dh) = d⇤�

(1� ↵�1)d⇤'�

.

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 75

Rewriting the left-hand side by

d⇤(↵�1dh) = �h� d⇤�

(1� ↵�1)dh� ⌘ T (h).

Letting W 1,p

? (M ;R) denote the subset of W 1,p(M ;R) consisting of functions integrat-

ing to zero, we see that the Hodge Laplacian, viewed as an operator from W 1,p

? (M ;R)to W�1,p(M ;R), has a bounded inverse. Recalling that |1 � ↵| �, we see that if

the parameter � in the definition of ↵ is small enough (cf. the start of Step 2 in

the proof of Theorem 2.18), then the operator T again has a bounded inverse from

W�1,p(M ;R) back into W 1,p

? (M ;R). Hence we infer that

(2.4.84) krhkp;M

CkT (h)k�1,p;M

Ckd⇤'kp;M

C.

Using the estimates (2.4.83) and (2.4.84) in the decomposition (2.4.82), we see that

ku⇥ dukp;M

C.

Step 2: Estimate for |r|u|2|2.We can follow the proof of Proposition VI.4 in [BBO01] word for word to get the

estimate we need. In fact, in our case the proof is easier since there is no boundary

term. Note that the domain ⌦ in [BBO01] should of course be replaced by our

manifold M . We would like to remind the reader of a few misprints in the proof given

in [BBO01]. Specifically, the definition of ⇢ at the end of page 487 should be

⇢ = max{⇢, 1� ✏1/2}, instead of max{⇢, 1� ⇢1/2},where ⇢ = |u|. Also, the integral on the left-hand side of the 5th line on page 488

should be ˆS

|r⇢|2, instead of

ˆM\S

|r⇢|2.

The estimate we end up obtaining is the following.

kr⇢kp;M

C✏1�p/2 |log ✏| .

Step 3: Conclusion

Given the Lp-estimates on u ⇥ du and r⇢ derived in the previous two steps, we

can conclude the proof of Proposition 2.40 exactly as in Section VI.4 of [BBO01].

Note that the Proposition 1 used there should be replaced by our Proposition 2.22.

This completes our sketch for the proof of Proposition 2.40. ⇤

With the help of the globalW 1,p-estimate (Proposition 2.40), we obtain the follow-

ing improvement of the ⌘-ellipticity theorem, which yields an energy bound uniform

in ✏. Starting from such a bound, we can bootstrap as in Proposition 2.31 and get

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 76

higher order estimates that will allow us to pass to a limit and get the desired smooth

harmonic map away from the set ⌃.

Proposition 2.41. Let g 2 M1,�,⇤

and suppose u : (B1

, g) ! D is a solution to

(2.1.2). Assume furthermore that for some p 2 [1, n/(n � 1)) there is a constant Cp

such that

(2.4.85) krukp;B1 C

p

.

Then there exists constants ⌘2

and ✏4

depending only on n,� and ⇤, such that if

(2.4.86)

ˆB1

e✏

(u) dvolg

⌘ |log ✏| , for some ⌘ < ⌘1

, ✏ < ✏2

,

then we have

(1) |u(x)| � 1/2 for x 2 B3/4

.

(2) The following energy bound holds.ˆB1/2

e✏

(u) dvolg

C,

where C depends on n,�,⇤, p and the constant Cp

.

(3) Restricting to a smaller ball, we have the following improvement of (2).

e✏

(u)(x) C

ˆB1/2

e✏

(u) dvolg

C, for x 2 B1/8

.

Remark 2.42. For convenience, we have stated this result in local form. Also, the

proofs of (1) and (2) are almost the same as Proposition VII.1 of [BBO01], but we

chose to include a short sketch of the arguments. Assertion (3) is a simple consequence

of (2) and Corollary 2.26.

Proof. Requirements on the constants ⌘2

and ✏4

will be specified as we go along. To

begin with, we demand that ⌘2

< ⌘1

and ✏4

< ✏2

, where ⌘1

and ✏2

are as in Corollary

2.26. Then, as the proof of that Corollary shows, we have |u| � 1� �0

> 1/2 on B3/4

(recall that �0

< 1/2). This proves (1).

Now, on the ball B3/4

we may introduce polar coordinates and write u = ⇢ei'.

Note that we may require that ' ⌘ �B3/4

' dvol 2 [0, 2⇡). Also, by the assumption

(2.4.85), the lower bound |u| � 1/2 on B3/4

, and the fact that ru = r⇢+ i⇢r', wehave

(2.4.87)1

2kr'k

p;B3/4+ kr⇢k

p;B3/4 C

p

.

We now recall the equations satisfied by ⇢ and ':

��⇢+ ⇢|r'|2 = ✏�2(1� ⇢2)⇢,(2.4.88)

div(⇢2r') = 0.(2.4.89)

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 77

Since 1 � ⇢ � 1/2 on B3/4

, we see that (2.4.89) is uniformly elliptic. Therefore we

can use the Cacciopolli (Reverse Poincare) inequality, the De Giorgi-Nash estimates

and the Poincare inequality to get

kr'k2;B9/16

Ck'� 'k2;B5/8

Ck'� 'kp;B3/4

(2.4.90)

Ckr'kp;3/4

C.

Next, we test the equation (2.4.88) against ⇣(1�⇢), where ⇣ is a cuto↵ function that’s

identically 1 on B1/2

and vanishes outside B9/16

. Then, integrating by parts and using

(2.4.90) and (2.4.85) to estimate the various terms obtained (cf. Proposition VII.1 in

[BBO01] for the details on this step), we arrive at the following estimate.

(2.4.91)

ˆB1/2

|r⇢|2 + (1� ⇢2)2

✏2dvol

g

C.

The proof of (2) is complete upon putting together this estimate and (2.4.90).

Finally, by requiring ⌘2

and ✏4

to be even smaller, we may apply Corollary 2.26

to B1/2

instead of B1

, obtaining

supx2B1/8

e✏

(u)(x) C

ˆB1/2

e✏

(u) dvolg

.

Combining this with (2), we get the estimate asserted in (3). The proof of the

Proposition is now complete. ⇤

Proof of Theorem 1.7(4). Now we are ready to finish the proof of Theorem 1.7(4).

To that end, let us take the set ⌃ and the subsequence {u✏

k

} to be as in Theorem

1.7(1)(2).

We begin by noticing that since M is simply-connected and thus supports no non-

zero harmonic 1-forms, the form must be identically zero. Therefore, for all x /2 ⌃,

there exists r > 0 such that µ(Br

(x)) = 0. Consequently, fixing a point x0

/2 ⌃ and

taking the constants ⌘2

and ✏4

from Proposition 2.41, we see that there exists a small

enough radius r1

and a threshold k0

such that for k � k0

we have

(|log ✏k

|� |log r1

|) � max{|log ✏k

| /2, |log ✏4

|},and that

r2�n

1

ˆB

r1 (x0)

e✏

k

(u✏

k

) dvol ⌘2

2|log ✏

k

| .

Note that r1

is fixed first depending solely on x0

, and then k0

is fixed depending on

x0

, r1

and ⌘2

.

Next, thanks to the above requirements and the usual rescaling process which we

have done a number of times already, we may assume that r1

= 1 and find Proposition

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 78

2.41 applicable, yielding, for k � k0

, that |u✏

k

| � 1/2 on B3/4

, and that

(2.4.92) supx2B1/8

e✏

k

(u✏

k

)(x) C,

with C independent of k. Then, increasing the threshold k0

if necessary, we invoke

Proposition 2.27 on B1/8

to see that

(2.4.93)

�

�

�

�

1� ⇢2k

✏2k

�

�

�

�

0;B1/16

C(1 + |ru|20;B1/8

) C,

where we’ve introduced polar coordinates on B1/8

and written u✏

k

= ⇢k

ei'k . Now,

as in Step 2 of the proof of Theorem 1.7(1)(2), we can use (2.4.93) along with the

multiplicative Schauder estimates to get

(2.4.94) |r⇢k

|0;B1/18

C✏k

.

At this point, we may use the estimates (2.4.92), (2.4.93) and (2.4.94) in place of

(2.4.27), (2.4.28) and (2.4.32), respectively, and repeat the proof of Proposition 2.31

to obtain the higher order estimates asserted there, but with no |log ✏k

|-term on the

right-hand sides. Specifically, for l = 0, 1, · · · we have

|r'k

|l,0;B2�2l�4

Cl

|Xk

|l,0;B2�2l�4

Cl

|r⇢k

|l,0;B2�2l�5

Cl

✏↵l

k

,

where Xk

= ✏�2

k

(1 � ⇢k

). Thus, taking a subsequence for each l and extracting a

diagonal subsequence, we may assume that 'k

and Xk

converge in C l(B2

�2l�5) for

l = 0, 1, · · · to some limits ' and X. By the definition of Xk

, we then see that

⇢k

converges to 1 in C l(B2

�2l�5) for l = 0, 1, · · · . Combining this with the equation

div(⇢2k

'k

) = 0, the limit ' of 'k

must be harmonic.

In conclusion, we’ve shown that uk

= ⇢k

ei'k converges in C l(B2

�2l�5) for all l � 0

to a limit map u = ei', where ' is a classical harmonic function in small enough balls.

However, this implies that u is an S1-valued harmonic map near the origin. Thus, we

have proved a local version of Theorem 1.7(4), and passage to the global version can

be achieved by a standard covering argument. ⇤

2.5. Convergence of Ginzburg-Landau solutions II: The singular part

In this section, we prove Theorem 1.7(3). As in [BBO01], an important ingredient

in the proof would be the theory of generalized varifolds, introduced by Ambrosio

and Soner [AS97]. However, since the Ambrosio-Soner theory was developed in the

Euclidean setting, it is not totally transparent how to apply it to the manifold case.

Another additional challenge we have in comparison with [BBO01] is the presence of

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 79

the harmonic 1-form . Therefore, much of this section will be devoted to resolving

these two issues and verifying that we can use Ambrosio-Soner’s result to produce

a stationary varifold. This will be done through a number of Lemmas, and we will

indicate how Theorem 1.7(3) follows from all these preparations at the end of this

section. To avoid confusion, we remind the reader that for the results in this section

we do not assume M is simply-connected.

To begin, let us recall that our goal is to show that the set ⌃ from the previous

section supports a stationary rectifiable (n�2)-varifold. The key would be the follow-

ing identity, proved in the same way (2.2.4) was derived in the proof of Proposition

2.10.

(2.5.1)

ˆM

�

e✏

k

(u✏

k

) divX � hrru

✏

k

X,ru✏

k

i� dvol = 0, for all C1-vector field X.

Below we again localize and consider a ball B ⌘ Br0(x0

), where x0

2 ⌃. On B we

introduce an orthonormal frame {ek

}nk=1

and write rk

for re

k

. Notice that we then

have divX =n

P

k=1

hrk

X, ek

i. Also, for notational convenience we will abbreviate u✏

k

and µ✏

k

simply as uk

and µk

, respectively. Finally, we will at times introduce polar

coordinates and write uk

= ⇢k

ei'k . The following lemma is a consequence of (2.4.1)

and the estimates in Proposition 2.31.

Lemma 2.43. There exists Radon measures {↵ij

}, supported on ⌃, such that the

following holds for all C1-vector field X with compact support in B.

0 =

ˆB

| |22

divX � hr

X, i dvol+ˆ⌃

divXd⌫ �ˆ⌃

hri

X, ej

id↵ij

,(2.5.2)

where in the second term of the first integral we viewed as a vector field.

Proof. The idea is to divide (2.5.1) by |log ✏k

| and pass to a limit. We first notice

that, by (2.4.1),

(2.5.3) limk!1

1

|log ✏k

|ˆB

e✏

k

(u✏

k

) divX dvol =

ˆB

| |22

divX dvol+

ˆ⌃

divXd⌫.

As for the second term in (2.5.1), we notice that, letting k

= |log ✏k

|�1/2 uk

⇥du

k

, we have k

! smoothly locally away from ⌃. Moreover, introducing polar

coordinates and writing uk

= ⇢k

ei'k , we see that

ri

uk

·rj

uk

= Re�r

i

urj

u�

(2.5.4)

= Re [(ri

⇢k

+ i⇢k

ri

'k

)(rj

⇢k

� i⇢k

rj

'k

)]

= ri

⇢k

rj

⇢k

+ ⇢2k

ri

'k

rj

'k

= ri

⇢k

rj

⇢k

+ |log ✏k

| ( k

)i

( k

)j

+ ⇢2k

(1� ⇢2k

)ri

'k

rj

'k

.

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 80

By the estimates in Proposition 2.31, we see that, in fact

(2.5.5) |log ✏k

|�1 ri

uk

·rj

uk

! i

j

, smoothy locally away from ⌃.

Now, since |log ✏k

|�1

´B

|ri

uk

||rj

uk

| dvol 2µk

(B) and the right-hand side is bounded

independent of k by (1.2.6), we can pass to a further subsequence and assume that

µij

⌘ limk!1

1

|log ✏k

|ri

uk

·rj

uk

dvol

exists as a Radon measure. We then see by (2.5.5) that µij

= i

j

dvol on M \⌃ and

that if we let

↵ij

= µij

� ( i

j

dvol),

then supp↵ij

⇢ ⌃. We now infer by the definition of ↵ij

that

(2.5.6)

limk!1

1

|log ✏k

|ˆB

hri

X,ruk

⌦ruk

i dvol =ˆB

hrX, ⌦ i dvol+ˆ⌃

hri

X, ej

id↵ij

.

Adding (2.5.3) and (2.5.6), we get the desired result. ⇤

Next we show that the harmonicity of implies that the first integral in (2.5.2)

always vanishes.

Lemma 2.44. For all C1-vector field X on M , we have

(2.5.7)

ˆM

| |22

divX � hr

X, i dvol = 0.

Proof. Let ' : [0, 1] ⇥M ! M denote the flow generated by the vector field X and

abbreviate '(t, ·) as 't

. We will use the follow version of the homotopy formula.

(2.5.8) '⇤1

� = d ['⇤ ](0,1)

+ ['⇤(d )](0,1)

,

where [·](0,1)

means to take the horizontal part (i.e. the components containing dt)

and integrate out the t-variable. We will omit the proof of (2.5.8) since it is a direct

consequence of the homotopy formula for currents. A proof of the latter can be found

in [Sim83], Chapter 6, Section 26.

Since is closed, we see from (2.5.8) applied to [0, t] instead of [0, 1] that '⇤t

is

cohomologous to for all t 2 [0, 1]. Now, since is harmonic and thus minimizes

the L2-norm in its cohomology class, we infer that

(2.5.9)

ˆM

|'⇤t

|2 dvol �ˆM

| |2 dvol, for all t 2 [0, 1].

Di↵erentiating in t at t = 0 gives (2.5.7) with � in place of =. Repeating the same

argument with �X in place of X, we get the desired equality in (2.5.7). ⇤

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 81

Combining the two previous Lemmas, we get

(2.5.10)

ˆ⌃

divXd⌫ �ˆ⌃

hri

X, ej

id↵ij

= 0,

for all C1-vector field X supported within B. We now take a closer look at each

integral. Note that, since ⌫ µ and µ satisfies (2.4.23), we easily see that

(2.5.11) lim supr!0

r2�n⌫(Br

) E0

for all x 2 ⌃.

Thus ⌫ is absolutely continuous with respect to Hn�2 on ⌃, and we can write

(2.5.12) ⌫ = ⇥(⌫, ·)Hn�2

x⌃.

Next, notice that by definition we have |µij

| 2µ, and that | i

j

dvol | | |2 dvol 2µ by (2.4.1). Thus, for x 2 ⌃ \B we have

r2�n↵ij

(Br

(x)) = r2�nµij

(Br

(x))� r2�n

ˆB

r

(x)

i

j

dvol

4r2�nµ(Br

(x)) 4E0

,

for r small enough, and we infer as above that ↵ij

is also absolutely continuous with

respect to Hn�2 on ⌃, and hence, using the fact that ⇥(⌫, x) � ⌘/4 > 0 for Hn�2-a.e.

x 2 ⌃, we can write

(2.5.13) ↵ij

= ⇥(⌫, ·)Aij

(·)Hn�2

x⌃ \ B(= Aij

(·)⌫xB).

Putting together (2.5.12) and (2.5.13), we see from (2.5.10) that

(2.5.14)

ˆ⌃

(�ij

� Aij

(x)) hri

X, ej

i⇥(⌫, x)dHn�2 = 0,

for all C1-vector fields X on M supported in B. Note that this equation resembles

a stationarity condition for a varifold, the problem being that we don’t know if the

matrices (�ij

�Aij

) are orthogonal projections onto (n� 2)-planes. Nonetheless, they

still define generalized varifolds in the sense of Ambrosio-Soner ([AS97]).

Lemma 2.45. For Hn�2-a.e. x 2 ⌃ \ B, the matrix I � A(x) ⌘ (�ij

� Aij

(x)) is

in the class A�1,n�2

(cf. [AS97], Definition 3.4), which consists of n ⇥ n-matrices

C = (Cij

) satisfying (1) tr(C) � n� 2 and (2) �I C I.

Remark 2.46. In fact, it is true that tr(I �A(x)) = n� 2 for Hn�2-a.e. x 2 ⌃\B.

However, just a lower bound would su�ce for our purposes.

Proof. Note that by the definition of ↵ij

we haven

X

i=1

↵ii

=n

X

i=1

µii

� | |2 dvol

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 82

= limk!1

1

|log ✏k

| |ruk

|2 dvol�| |2 dvol

= limk!1

2µk

� limk!1

1

|log ✏k

|(1� |u

k

|2)24✏2

k

dvol�| |2 dvol

2µ� | |2 dvol = 2⌫.

Therefore we infer using (2.5.13) and Lesbesgue’s di↵erentiation theorem that trA(x) 2 for Hn�2-a.e. x 2 ⌃ \ B and hence tr(I � A(x)) � n� 2 for Hn�2-a.e. x 2 ⌃ \ B.

This proves (1).

To see that (2) holds, we take an arbitrary ⇠ 2 Rn and observe that

⇠i

⇠j

↵ij

= ⇠i

⇠j

µij

� h , ⇠i2 dvol = limk!1

1

|log ✏k

|hruk

, ⇠i2 dvol�h , ⇠i2 dvol .

Now, by (2.5.5) and Proposition 2.31 we have

1

|log ✏k

|hruk

, ⇠i2 ! h , ⇠i2 smoothly locally o↵ of ⌃,

and hence we infer that ⇠i

⇠j

↵ij

� 0 and hence A(x) is positive-definite for Hn�2-a.e.

x 2 ⌃ \ B. This proves the second inequality in (2).

To get the first inequality in (1) we need an upper bound on A. For this we note

that

limk!1

1

|log ✏k

|hruk

, ⇠i2 dvol�h , ⇠i2 dvol 2µ|⇠|2.

Therefore, by Lebesgue’s di↵erentiation theorem, for µ-a.e. x 2 ⌃ \B we have

lim⇢!0

⇠i

⇠j

↵ij

(B⇢

(x))

µ(B⇢

(x)) 2|⇠|2.

Note that the above inequality then holds Hn�2-a.e. x 2 ⌃ by the lower density

bound in (2.4.23). Now, recalling (2.4.68) from the previous section, we see that

(2.5.15) lim⇢!0

µ(B⇢

(x))

⌫(B⇢

(x))= 1 for Hn�2-a.e.x 2 ⌃.

Combining the two inequalities above, we conclude that ⇠i

⇠j

Aij

(x) 2|⇠|2 and hence

I � A(x) � �I for Hn�2-a.e. x 2 ⌃ \ B. This proves the first inequality in (2). The

proof of the Lemma is now complete. ⇤

Since we are working on Riemannian manifolds, one more step is needed to put

ourselves in position to apply the theory of generalized varifolds developed in [AS97].

To that end, we will assume that our manifold M is isometrically embedded in an

Euclidean space RN . Note that by the smoothness of M , the densities ⇥(µ, ·) and

⇥(⌫, ·) remain the same regardless of whether we compute them use geodesic balls

or ambient balls. Also, denoting ambient balls by BN

r

(x), we will with out loss of

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 83

generality that BN

⇢/2

(x) ⇢ B⇢

(x) ⇢ BN

2⇢

(x) for all x 2 B(= Br0(x0

)) and ⇢ < r0

. Note

that this can always be achieved by taking r0

small enough.

The following result provides the bridge between our situation and that considered

in [AS97]. For convenience, we will denote �ij

� Aij

(·) by Sij

(·) and ⇥(⌫, ·) by ✓(·).Lemma 2.47. For Hn�2-a.e. x 2 ⌃\B, the following holds for all X 2 C1

0

(BN

1

;RN).

(2.5.16) Sij

(x)

ˆB

N

1 \Tx

⌃

hre

i

(x)

X(z), ej

(x)idHn�2(z) = 0,

where Tx

⌃ denotes the approximate tangent plane to ⌃ at x.

Remark 2.48. Note that X 2 C1

c

(BN

1

;RN) is now an ambient vector field and is not

required to be tangent to M . Also, recall that {ek

(x)}nk=1

is an orthonormal frame

for Tx

M for x 2 B.

Proof of Lemma 2.47. In short, the proof is accomplished through rescaling the iden-

tity (2.5.14) and noticing that the extra term introduced by the 2nd fundamental

form of M scales away in the process. The details are as follows.

For any vector field X 2 C1

0

(B;RN), we break it up as X = X t + Xn, where X t

is tangent to M and Xn normal to M . Then, by (2.5.14) we haveˆB

Sij

hri

X, ej

i✓(·)dHn�2 =

ˆB

Sij

hri

Xn, ej

i✓(·)dHn�2(2.5.17)

= �ˆB

Sij

hHij

, Xi✓(·)dHn�2,

where we let Hij

= rni

ej

denote the second fundamental form of M in RN .

Now let X be a vector field as in the statement of the Lemma. Taking a point

x 2 ⌃ \ B, for ⇢ < (r0

� |x0

� x|)/2 we define

(2.5.18) X(y) = X

✓

y � x

⇢

◆

.

Then X is supported within BN

⇢

(x) ⇢ B2⇢

(x) ⇢ B, and we can use (2.5.17) along

with the area formula to see that

� ⇢3�n

ˆB

N

⇢

(x)

Sij

✓(y)hHij

(y), Xn(y)idHn�2(y)(2.5.19)

=

ˆB

Sij

(x+ ⇢z)✓(x+ ⇢z)hre

i

(x+⇢z)

X(z), ej

(x+ ⇢z)idHn�2(z).

Note that, by Lemma 2.45(2), the definition of X, and the remarks preceding Lemma

2.47, the left-hand side of (2.5.19) is bounded above by

C|Hij

|0;M

|X|0;B

N

1⇢3�n

ˆB2⇢(x)

✓(y)dHn�2(y)

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 84

C|X|0;B

N

1⇢�

⇢2�n⌫(B2⇢

(x))�

,

where we absorbed the 2nd fundamental form bound into the constant. Choosing

now x to be a point where ⇥(⌫, x) E0

, we see that, as ⇢ goes to zero,

(2.5.20) � ⇢3�n

ˆB

N

⇢

(x)

Sij

(y)✓(y)hHij

(y), Xn(y)idHn�2(y) ! 0.

This takes care of the left-hand side of (2.5.19). Next we treat the right-hand side.

Claim. Up to a multiplicative constant depending only on n, for Hn�2-a.e. x 2 ⌃\Bthe right-hand side of (2.5.19) tends to

(2.5.21) Sij

(x)✓(x)

ˆB

N

1 \Tx

⌃

hre

i

(x)

X(z), ej

(x)idHn�2(z), as ⇢! 0,

where Tx

⌃ denotes the approximate tangent plane to ⌃ at x.

Proof of the claim. We notice that we can writeˆB

N

1

Sij

(x+ ⇢z)✓(x+ ⇢z)hre

i

(x+⇢z)

X(z), ej

(x+ ⇢z)idHn�2(z)

=

ˆB

N

1

Sij

(x+ ⇢z)✓(x+ ⇢z)⇥hr

e

i

(x+⇢z)

X(z), ej

(x+ ⇢z)i � hre

i

(x)

X(z), ej

(x)i⇤ dHn�2(z)

+

ˆB

N

1

[Sij

(x+ ⇢z)� Sij

(x)] ✓(x+ ⇢z)hre

i

(x)

X(z), ej

(x)idHn�2(z)

+ Sij

(x)

ˆB

N

1

hre

i

(x)

X(z), ej

(x)i✓(x+ ⇢z)dHn�2(z)

⌘ I + II + III.

We shall see that, as ⇢ ! 0, (I) and (II) tends to zero while (III) tends to the

expression in the statement of the Claim.

To estimate (I), we simply notice that since the frame {ek

} is smooth and X is

C1, and since each Sij

is essentially bounded with respect to Hn�2

x⌃ by Lemma 2.45,

we have

|(I)| C⇢

ˆB

N

1

✓(x+ ⇢z)dHn�2(z) C⇢⇢2�n⌫(B2⇢

(x)) C⇢E0

! 0 as ⇢! 0,

where in the last inequality we used the density upper bound in (2.4.2), which holds

for Hn�2-a.e. x 2 ⌃ \ B.

Next, for (II), we estimate

|(II)| C|rX|0;B

N

1⇢2�n

ˆB

N

⇢

(x)

|Sij

(y)� Sij

(x)| ✓(y)dHn�2(y)

CE0

|rX|0;B

N

1⇢2�n

ˆB

N

⇢

(x)

|Sij

(y)� Sij

(x)| d(Hn�2

x⌃)(y).

2. GINZBURG-LANDAU CRITICAL POINTS ON CLOSED MANIFOLDS 85

Note that the expression in the last line tends to zero as ⇢ ! 0 if we make the

following requirements: (1) x is a Lebesgue point of each Sij

with respect to Hn�2

x⌃;

(2) lim⇢!0

⇢2�nHn�2(BN

⇢

(x)\⌃) exists. Note that since Sij

is L1 with respect to Hn�2

x⌃

and since ⌃ is rectifiable, both conditions hold for Hn�2-a.e. x 2 ⌃ \B.

Finally, (III) tends to the desired expression by the definition of an approximate

tangent plane, provided we assume that the approximate tangent plane to ⌃ exists

at x. Since all the requirements we made on x hold Hn�2-a.e on ⌃ \ B, the claim is

proved. ⇤

By (2.5.19), (2.5.20) and the above Claim, we’ve shown that for a fixed X 2C1

0

(BN

1

;RN), the following holds for Hn�2-a.e. x 2 ⌃ \B,

(2.5.22) Sij

(x)✓(x)

ˆB

N

1 \Tx

⌃

hre

i

(x)

X(z), ej

(x)idHn�2(z) = 0.

The lower density bound in (2.4.2) allows us to drop ✓(x) from the left-hand side for

Hn�2-a.e. x 2 ⌃ \ B. Finally, by a density argument we find a set with Hn�2-full

measure in ⌃ \ B where (2.5.16) holds for all X 2 C1

0

(BN

1

;RN). ⇤

For later purposes, we fix t 2 (0, 1) satisfying t < n�1/2 and t�1 > nn/2, and

augment the matrices S(x) to become N ⇥N by putting S(x) in the top-left n⇥ n-

block and letting all other entries be zero. We will still denote the augmented matrices

by S(x).

Taking the above choice of t, we are now ready to apply Lemma 3.9 of [AS97].

We will replace the measure ⌫ there by !�1

n�2

Hn�2

xTx

⌃, where !n�2

is the volume of

the unit (n�2)-ball. Moreover, the dimension d should be replaced by N , the powers

�, s should both be replaced by n� 2, and the matrix A should be replaced by S(x),

where x is one of the Hn�2-a.e. points where (2.5.16) hold for all X 2 C1

0

(BN

1

;RN).

Then the conclusion of Lemma 3.9 implies that at least N �n+2 eigenvalues of S(x)

are 0.

Now, by condition (2) in Lemma 2.45, all N of the eigenvalues of S(x) lie in [�1, 1],

and by condition (1) they sum up to at least n� 2 (the augmentation we performed

on S(x) does not change the conclusions of Lemma 2.45). Since we already know that

at least N � n+ 2 of the eigenvalues are 0, the only way both conditions can be met

is if exactly N �n+2 of the eigenvalues are 0, while the remaining n� 2 eigenvalues

are all 1. Therefore, we conclude that for Hn�2-a.e. x 2 ⌃ \ B, the matrix S(x) is

the orthogonal projection onto an (n� 2)-dimensional subspace. Now, the following

Lemma provides the last piece in the puzzle for the proof of Theorem 1.7(3).

Lemma 2.49. Fix an x where (2.5.16) holds. Then S(x) projects onto Tx

⌃ for

Hn�2-a.e. x 2 ⌃ \B.

3. EXISTENCE OF STATIONARY RECTIFIABLE VARIFOLDS 86

Proof of Lemma 2.49. For this proof we decompose RN into Tx

⌃� T?x

⌃� T?x

M and

write a point z 2 RN as z = (z0, z00, y) = (z1

, · · · , zn�2

, zn�1

, zn

, y1

· · · , yN�n

) accord-

ing to this decomposition. Note that in doing so we have assumed that ek

(x) = @

@z

k

for k = 1, · · · , n. Without loss of generality, we may further assume that {ek

(x)}n�2

k=1

forms an orthonormal basis for Tx

⌃.

From the definition of S(x), it is clear that T?x

M lies in its kernel. Therefore it

remains to show that S(x) annihilates en�1

and en

as well. To that end let us fix

k 2 {1, · · · , N}, ⇣ 2 C1

0

(BN

1

) and substitute X = ⇣ek

into (2.5.16). Then we get,

(2.5.23) Ski

(x)

ˆT

x

⌃

re

i

(x)

⇣dHn�2 = 0.

We now take j 2 {n� 1, n} and make the following choice of ⇣.

(2.5.24) ⇣(z0, z00, y) = ⇠(z0)zj

, where ⇠ 2 C1

0

(Bn�2

1

).

Note that since the integral in (2.5.23) is over Tx

⌃, the above is a legitimate choice of

⇣ even though it does not have compact support in the (z00, y)-direction. Substituting

this ⇣ into (2.5.23) we obtain

Ski

(x)

ˆT

x

⌃

@

@zi

⇠(z0)zj

+ ⇠(z0)�ij

dHn�2(z) = 0.

Notice that by our choice of j, the first term in the integrand vanishes on Tx

⌃ (which

we are assuming to be Rn�2 ⇥ {0}) and hence we infer that

Skj

(x)

ˆT

x

⌃

⇠(z0)dHn�2(z) = 0, for all ⇠ 2 C1

0

(Bn�2

1

),

and therefore Skj

(x) = 0. Recalling that k is arbitrary and that S(x) is in fact

symmetric, we conclude that S(x)ej

= 0 for j = n� 1, n. In summary, we’ve shown

that the kernel of the projection S(x) is T?x

⌃�T?x

M and hence it projects onto Tx

⌃.

The Lemma is proved. ⇤

In view of the above Lemma, the remarks preceding it, and the identity (2.5.14),

we conclude that the pair (⌃,⇥(⌫, ·)) defines a stationary rectifiable varifold in B =

Br0(x0

). Since x0

2 ⌃ is arbitrary, we conclude that Theorem 1.7(3) is proved.

3. Existence of stationary rectifiable varifolds

The content of Sections 3.1 to 3.3 reflects our e↵ort toward using Theorem 1.5 to

establish the existence of stationary rectifiable (n � 2)-varifolds in a closed, simply-

connected Riemannian manifold M . To be precise, in Section 3.1 we introduce a

modified version of the Ginzburg-Landau functional and discuss its important vari-

ational properties on W 1,2(M ;C). Then, in Section 3.2, we apply the saddle point

3. EXISTENCE OF STATIONARY RECTIFIABLE VARIFOLDS 87

theorem of Rabinowitz ([Rab86]) to obtain a sequence of non-constant Ginzburg-

Landau critical points {u✏

} which we expect to satisfy the assumption (1.2.6) in

Theorem 1.5. This last assertion is partially verified in Section 3.3, where we show

that {u✏

} indeed satisfies the lower bound in (1.2.6).

3.1. The modified Ginzburg-Landau functional

In this section we prove that a suitably modified version of the Ginzburg-Landau

functionals satisfy a variant of the Palais-Smale compactness condition ([Pal63],

[Sma64]), which would allow us to apply standard min-max theory to produce a

sequence of non-trivial critical points.

Since we would like to work in the Sobolev space W 1,2(M ;C), the quartic growthof the second term of e

✏

(u) poses a problem. To go around that, we modify the energy

density e✏

(u) by letting

(3.1.1) e✏

(u) =|ru|22

+W (|u|)✏2

,

where the potential W : [0,1) ! [0,1) is given by

(3.1.2) W (t) =

(

(t

2�1)

2

4

, t 1,

(t� 1)2, t > 1.

Observe thatW is still a C2 function on [0,1). We then define the modified Ginzburg-

Landau energy as

E✏

(u) =

ˆM

e✏

(u) dvol, u 2 W 1,2(M ;C).

Before proceeding, we collect in the Lemma below some important properties of

the potential W that we will frequently use in analyzing E✏

.

Lemma 3.1. (1) W 0(t)/t 2 for all t � 0.

(2) W 00(t) 2 for all t � 0.

(3) Let z, w 2 C ' R2. Then we have

(3.1.3)

�

�

�

�

W 0(|z|)✏2

z

|z| �W 0(|w|)

✏2w

|w|�

�

�

�

6

✏2|z � w|.

Proof. For (1) we notice that

(3.1.4)

�

�

�

�

W 0(t)

t

�

�

�

�

=

(

1� t2, t 1,

2� 2t�1 2, t > 1.

Hence the asserted inequality follows immediately. (2) is proved in a similar way.

For (3), we introduce the notation zt

= tz + (1� t)w (0 t 1) and write

(3.1.5)W 0(|z|)✏2

z

|z| �W 0(|w|)

✏2w

|w| =ˆ

1

0

d

dt

✓

W 0(|zt

|)✏2

zt

|zt

|◆

dt.

3. EXISTENCE OF STATIONARY RECTIFIABLE VARIFOLDS 88

A direct computation shows that the integrand is equal to

d

dt

✓

W 0(|zt

|)✏2

zt

|zt

|◆

=W 00(|z

t

|)✏2

zt

· (z � w)

|zt

|2 zt

+W 0(|z

t

|)✏2

z � w

|zt

| � W 0(|zt

|)✏2

zt

· (z � w)

|zt

|3 zt

.

Using assertions (1) and (2) we see that each of the three summands above is bounded

in absolute value by 2

✏

2 |z � w|. Thus, putting this back into (3.1.5), we obtain

(3.1.6)

�

�

�

�

W 0(|z|)✏2

z

|z| �W 0(|w|)

✏2w

|w|�

�

�

�

ˆ

1

0

6

✏2|z � w|dt = 6

✏2|z � w|.

The proof of the Lemma is complete. ⇤

Note that since W has quadratic growth at infinity, E✏

is well-defined for functions

in W 1,2(M ;C). This modification is further justified by the next proposition, which

shows that the critical points of E✏

in W 1,2(M ;C) are in fact smooth solutions to the

Ginzburg-Landau equation (2.1.2).

Proposition 3.2. Let u be a critical point of E✏

in the sense that

(3.1.7)

ˆM

hru,r⇣i+ W 0(|u|)✏2

u · ⇣|u| dvol = 0, for all ⇣ 2 W 1,2(M ;C).

Then |u| 1 and u is a smooth solution to (2.1.2).

Proof. We first show that u is C2,↵ everywhere on M for some ↵ 2 (0, 1). Note that

by (3.1.7), u is a weak solution to

(3.1.8) �u =W 0(|u|)✏2

u

|u| .

Notice that (3.1.8) can be rewritten as

(3.1.9) �u� W 0(|u|)✏2|u| u = 0

By Lemma 3.1(1), the coe�cient of u lies in L1. Therefore, Schauder theory gives

u 2 C1,↵(M ;C) for some ↵ > 0.

Next, on the open set {|u| < 1}, we see that, by the definition of W , the equation

(3.1.8) reduces to (2.1.2), and hence u is smooth there. On the other hand, on

{|u| > 1/2}, since u 2 C1,↵(M ;C), the right-hand side of (3.1.8) is in C0,↵. Thus u

is C2,↵ there.

Now that we’ve shown that u 2 C2,↵ everywhere, we proceed to prove that |u| 1

on M . To that end, we compute

(3.1.10) �|u|2 = 2|ru|2 + 2W 0(|u|)✏2

|u| �(

2

✏

2 |u|2(|u|2 � 1), |u| 1,4

✏

2 |u|(|u|� 1), |u| > 1.

3. EXISTENCE OF STATIONARY RECTIFIABLE VARIFOLDS 89

Suppose {|u| > 1} 6= ;, then by the above computation, on this open set we have

�|u|2 � 0. The maximum principle then implies that |u|2 is either constant on

{|u| > 1} or attains its maximum there on @{|u| > 1}. In both cases, we get |u| 1

in {|u| > 1}, a contradiction, and therefore {|u| > 1} = ;; in other words, |u| 1

everywhere on M . Consequently, u satisfies (2.1.2) on M , and is therefore smooth by

Proposition 2.1, for instance. ⇤

We next verify a version of the Palais-Smale condition for E✏

for all ✏ > 0. The

definition of this condition will be recalled in the statement of the following Proposi-

tion.

Proposition 3.3. Let {ui

} be a sequence in W 1,2(M ;C) satisfying(1) sup

i

E✏

(ui

) C0

< 1.

(2) limi!1

k�E✏

(ui

)k = 0, where �E✏

is the first variation operator defined by

(3.1.11) �E✏

(u)(v) =

ˆM

hru,rvi+ W 0(|u|)✏2

u · v|u| dvol,

and the k · k-norm is defined by

(3.1.12) k�E✏

(u)k = sup{�E✏

(u)(v)| v 2 W 1,2(M ;C), kvk1,2;M

1}.Then, passing to a subsequence if necessary, {u

i

} converges strongly in W 1,2(M ;C)to a critical point u of E

✏

. In other words, E✏

satisfies the Palais-Smale condition on

W 1,2(M ;C).

Remark 3.4. The version of the Palais-Smale condition we use above is taken from

[Str08]. We note that it is slightly stronger than the original version, which was

introduced as ”Condition (C)” in [Pal63].

Proof of Proposition 3.3. We first see that assumption (1) and the definition of E✏

that krui

k22;M

2C0

. Moreover, by the definition of W we haveˆM\{|u|�1}

(|u|� 1)2

✏2 C

0

,

and hence kui

k2;M

is uniformly bounded as well. Hence we’ve shown that {ui

} is

a bounded sequence in W 1,2(M ;C) and we may assume that it converges weakly in

W 1,2(M ;C), strongly in L2(M ;C) and pointwise a.e. to some u.

By the convergence of {ui

} to u, Lemma 3.1(1), and the dominated convergence

theorem, we infer that for all v 2 W 1,2(M ;C),

�E✏

(u)(v) = limi!1

�E✏

(ui

)(v) lim supi!1

k�E✏

(ui

)kkvk1,2;M

! 0 as i ! 1,

and hence u is a critical point of E✏

.

3. EXISTENCE OF STATIONARY RECTIFIABLE VARIFOLDS 90

Next we use assumption (2) to upgrade the weak W 1,2-convergence of {ui

} to

strong W 1,2-convergence. To do that, we take i < j and use (3.1.11) with u = ui

and

v = ui

� uj

. Then we notice that, by (3.1.12)

(3.1.13)�

�

�

�E✏

(ui

)(ui

� uj

)� �E✏

(uj

)(ui

� uj

)�

�

�

⇣

k�E✏

(ui

)k+ k�E✏

(uj

)k⌘

kui

� uj

k1,2;M

.

Since {ui

} is bounded in W 1,2(M ;C), by condition (2) we infer that the right-hand

side in the above expression tends to zero as i, j ! 1. We now compute the left-hand

side.

�E✏

(ui

)(ui

� uj

)� �E✏

(uj

)(ui

� uj

)(3.1.14)

=

ˆM

|r(ui

� uj

)|2 +✓

W 0(|ui

|)✏2

ui

|ui

| �W 0(|u

j

|)✏2

uj

|uj

|◆

· (ui

� uj

) dvol .

Recalling that W 0(t)/t is bounded by 2 everywhere, we infer by the strong L2-

convergence of {ui

} thatˆM

�

�

�

�

✓

W 0(|ui

|)✏2

ui

|ui

| �W 0(|u

j

|)✏2

uj

|uj

|◆

· (ui

� uj

)

�

�

�

�

dvol

ˆM

2

✏2(|u

i

|+ |uj

|)|ui

� uj

| dvol ! 0 as i, j ! 1.

Using this along with (3.1.14) and recalling that the left-hand side of (3.1.14) tends

to zero as i, j ! 0, we conclude that

(3.1.15) limi,j!1

ˆM

|rui

�ruj

|2 dvol = 0.

Therefore {ui

} converges strongly in W 1,2(M ;C). ⇤

In order to apply standard results from critical point theory, we will also need

to show that E✏

is a C1-functional on W 1,2(M ;C). Recall that, for a Banach space

X, a functional E : X ! R is C1 if it is continuous and its Frechet derivative, as

a mapping from X to L(X,X 0), is continuous. Here X 0 denotes the dual of X, and

L(X,X 0) denotes the set of bounded linear maps from X into X 0. Below we verify

this for E = E✏

and X = W 1,2(M ;C).

Proposition 3.5. The functional E✏

: W 1,2(M ;C) ! R is C1.

Proof. Using Lemma 3.1(1) and the dominated convergence theorem, we see at once

that E✏

is continuous on W 1,2(M ;C). Thus we will focus on proving the continuity

of its Frechet derivative, which we denote by E 0✏

.

We first show that E 0✏

coincides with the first variation operator �E✏

. To see this,

we need to show that

(3.1.16) E✏

(u+ v)� E✏

(u) = �E✏

(u)(v) + o(kvk1,2;M

) as kvk1,2;M

! 0.

3. EXISTENCE OF STATIONARY RECTIFIABLE VARIFOLDS 91

A direct computation shows that

E✏

(u+ v)� E✏

(u) =

ˆM

hru,rvi+ |rv|22

+W (|u+ v|)�W (|u|)

✏2dvol

Fix a point x 2 M . By the mean value theorem, we can write, for some ✓ 2 (0, 1),

the following.

W (|u(x) + v(x)|)�W (|u(x)|)✏2

=W 0(|u(x) + ✓v(x)|)

✏2u(x) + ✓v(x)

|u(x) + ✓v(x)| · v(x)

=W 0(|u(x)|)

✏2u(x)

|u(x)| · v(x)

+

W 0(|u(x) + ✓v(x)|)✏2

u(x) + ✓v(x)

|u(x) + ✓v(x)| · v(x)�W 0(|u(x)|)

✏2u(x)

|u(x)| · v(x)�

.

By Lemma 3.1(3) with z = u(x)+✓v(x) and w = u(x), we see that the term in square

brackets is bounded in absolute value by

6

✏2✓|v(x)|2 6

✏2|v(x)|2

Therefore we get, for all x 2 M ,�

�

�

�

W (|u(x) + v(x)|)�W (|u(x)|)✏2

� W 0(|u(x)|)✏2

u(x)

|u(x)| · v(x)�

�

�

�

6

✏2|v(x)|2.

Plugging this back into (3.1.16) and recalling the definition of �E✏

, we obtain�

�

�

E✏

(u+ v)� E✏

(u)� �E✏

(u)(v)�

�

�

ˆM

|rv|22

+6

✏2|v|2 dvol 6

✏2kvk2

1,2;M

= o(kvk1,2;M

) as kvk1,2;M

! 0.

Hence �E✏

coincides with the Frechet derivative of E✏

.

To see that �E✏

: W 1,2(M ;C) ! (W 1,2(M ;C))0 is continuous, notice that, for u, uand v in W 1,2(M ;C),

�

�

�

�E✏

(u)(v)� �E✏

(u)(v)�

�

�

ˆM

|hru�ru,rvi|+�

�

�

�

W 0(|u|)✏2

u

|u| �W 0(|u|)✏2

u

|u|�

�

�

�

|v| dvol

kru�ruk2;M

krvk2;M

+

ˆM

6

✏2|u� u||v| dvol

6

✏2ku� uk

1,2;M

krvk2;M

,

3. EXISTENCE OF STATIONARY RECTIFIABLE VARIFOLDS 92

where in the second inequality we used Lemma 3.1(1) with z = u and w = u. Hence

we’ve shown that

(3.1.17) k�E✏

(u)� �E✏

(u)k 6

✏2ku� uk

1,2;M

,

and thus �E✏

is continuous. ⇤

3.2. Existence of min-max Ginzburg-Landau critical points

We recall the saddle point theorem of Rabinowitz ([Rab86]).

Theorem 3.6 (Rabinowitz, [Rab86], Theorem 4.6). Let X be a Banach space with

a decomposition X = V � Y , where 0 < dim(V ) < 1. Suppose E : X ! R is a

C1-functional satisfying the Palais-Smale condition, with the following properties.

(1) There exists a constant ↵ and bounded neighborhood D ⇢ V of the origin such

that E(u) ↵ for all u 2 @D.

(2) There exists a constant � > ↵ such that E(u) � � for all u 2 Y .

Then E has a critical point u with E(u) = c � �. Moreover, the critical value c can

be identified through the following min-max procedure.

(3.2.1) c = infh2�

maxu2 ¯

D

E(h(u)),

where � is the family defined by

(3.2.2) � = {h 2 C0(D;X)| h = id on @D}.Below we will apply Theorem 3.6 with X = W 1,2(M ;C) and E = E

✏

. Note that

we have verified in the previous section that E✏

is a C1-functional on W 1,2(M ;C)satisfying the Palais-Smale condition. Thus it remains to find subspaces V, Y , a

neighborhood D and numbers ↵, � satisfying assumptions (1) and (2) of Theorem

3.6.

The choice we make for Y is

(3.2.3) Y =

⇢

u 2 W 1,2(M ;C)|ˆM

u dvol = 0

�

,

and we let V be the collection of constant functions from M to C. Obviously, V = Y ?

and we have the decomposition W 1,2(M ;C) = V � Y . Furthermore, we take D to be

(3.2.4) D = {u 2 V | u ⌘ a 2 D},where D is the unit disk in C. Since W (1) = 0, it is obvious that E

✏

(u) = 0 whenever

u 2 @D. Thus we’ve verified assumption (1) with ↵ = 0.

As for assumption (2), notice that for u 2 Y , we have by the Poincare inequality

(3.2.5) E✏

(u) �ˆM

c|u|2 + W (|u|)✏2

dvol,

3. EXISTENCE OF STATIONARY RECTIFIABLE VARIFOLDS 93

where c depends only on n and M . To proceed, we observe that

(3.2.6) ct2 +W (t)

✏2�(

c/4, t � 1/2,

1/(8✏2), t < 1/2.

Combining the above inequalities, we see that when u 2 Y and ✏ < 1,

(3.2.7) E✏

(u) � min{c/4, 1/8}Hn(M) ⌘ � > 0 (= ↵).

We can now apply Theorem 3.6 to get the existence of a critical point u✏

of E✏

, whose

energy is given by

(3.2.8) E✏

(u✏

) ⌘ c✏

⌘ infh2�

maxu2 ¯

D

E✏

(h(u)).

Note that by Proposition 3.2, we infer that |u✏

| 1 pointwise on M , that E✏

(u✏

) =

E✏

(u✏

), and that u✏

is a smooth solution to (2.1.2). Also, since c✏

� � > 0 by Theorem

3.6, we conclude that each u✏

is non-constant.

In order to apply Theorem 1.5 and produce a non-zero stationary rectifiable (n�2)-

varifold in simply-connected ambient manifolds, it remains to show that the sequence

{u✏

} satisfies the upper bound and lower bound in (1.2.6). Unfortunately, we were

only able to obtain the lower bound. This is the content of Theorem 1.9, the proof

of which will be given in the next section.

3.3. Energy estimates for Ginzburg-Landau critical points: The lower

bound

Since we have constructed a sequence {u✏

} with each u✏

being a non-constant solu-

tion to (2.1.2), it remains to establish the following result in order to prove Theroem

1.9.

Proposition 3.7. Assuming that M is simply-connected, there exists a positive con-

stant � independent of ✏ such that the sequence {u✏

} obtained in the previous section

verifies the bound

(3.3.1) lim inf✏!0

E✏

(u✏

)

|log ✏| � �.

Proof. We will argue by contradiction. Suppose (3.3.1) is false, then there would exist

a subsequence {u✏

k

} such that

(3.3.2) limk!1

E✏

k

(u✏

k

)

|log ✏k

| = 0.

However, this implies that the limit measure µ is zero, so that, by definition, ⌃ =

;. Combining this last fact with Theorem 1.7(4), we infer that, up to taking a

subsequence, u✏

k

converges smoothly on M to a smooth harmonic map u : M ! S1.

Since M is simply-connected, we may write u = ei', but then the fact that u is

3. EXISTENCE OF STATIONARY RECTIFIABLE VARIFOLDS 94

a harmonic map implies that ' is a harmonic function on M , and hence must be

constant since @M = ;. Thus, we conclude that u ⌘ a for some a 2 @D.On the other hand, by construction, we have

(3.3.3) E✏

k

(u✏

k

) � � > 0,

where � is independent of k. We will derive a contradiction using this lower bound.

Specifically, by the smooth convergence of {u✏

k

} and the harmonic map equation, we

see that

(3.3.4)1� |u

✏

k

|2✏2k

u✏

k

= ��u✏

k

! ��u = |ru|2u uniformly as k ! 1,

and hence Xk

⌘ ✏�2

k

(1 � |u✏

k

|2) ! |ru|2 uniformly as k ! 1. This in turn implies

that

(3.3.5)(1� |u

✏

k

|2)2✏2k

= ✏2k

X2

k

! 0 uniformly as k ! 1.

Therefore the potential part of E✏

k

(u✏

k

) is negligible in the limit as k goes to infinity,

and we infer that

(3.3.6) lim infk!1

1

2kru

✏

k

k22;M

= lim infk!1

E✏

k

(u✏

k

) � � > 0.

In particular, the limit map u must be non-constant, which is in contradiction with

what we showed in the last paragraph. The proof of Proposition 3.7, and hence of

Theorem 1.9, is now complete. ⇤

Remark 3.8. As mentioned in the Introduction, Stern [Ste16], independently, used

a method similar to ours to obtain the sequence {u✏

} and has managed to verify both

the upper bound and the lower bound.

APPENDIX A

The fundamental solution of divergence-form elliptic systems

The purpose of this appendix is to derive estimates on the fundamental solution

for elliptic systems in divergence form. We shall see that under mild assumptions on

the coe�cients, the fundamental solution has the same growth near the diagonal as

in the constant-coe�cient case. This estimate is crucial to our proof of Theorem 2.18.

We will be interested in elliptic operators of the following type:

(A.1) (Lu)↵

⌘ @i

�

Aij

↵�

@j

u�

+Bi

↵�

u�

�

+ C i

↵�

@i

u�

+D↵�

u�

(↵ = 1, · · · , N),

where u = (u1

, · · · , uN

) maps B1

⇢ Rn into RN , and the repeated indices are meant

to be summed. The Latin indices are understood to run from 1 to n, while the Greek

indices go from 1 to N . The assumptions on the coe�cients of L are as follows. For

some fixed constants ⇤, � > 0, and 1 � µ > 0, we assume that

(H1) Aij

↵�

2 C0,µ(B1

) and Bi

↵�

, Ci

↵,�

, D↵,�

2 L1(B1

) for all i, j,↵, �.

(H2) [Aij

↵�

]0,µ;B1 + kBi

↵�

k1;B1 + kC i

↵�

k1;B1 + kD↵�

k1;B1 ⇤, for all i, j,↵, �.

(H3) �|⇠|2 Aij

↵�

(x)⇠i↵

⇠j�

��1|⇠|2, for all x 2 B1

and ⇠ = (⇠i↵

) in RnN .

Let c0

be another constant. Later in this appendix, we will also impose the following

coercivity condition:

(H4) For all u 2 W 1,2

0

(B1

;RN), there holds

(A.2)

ˆB1

�

Aij

↵�

@j

u�

+Bi

↵�

u�

�

@i

u↵

� u↵

�

C i

↵�

@i

u�

+D↵�

� � c0

kruk22;B1

,

We recall that systems involving Lu should be understood in the weak sense.

Specifically, given F = (F i

↵

) 2 L1

loc

(B1

;RnN) and f = (f↵

) 2 L1

loc

(B1

;RN), we say

that u 2 W 1,1

loc

(B1

;RN) is a (weak) solution on B1

to

(A.3) (Lu)↵

= @i

F i

↵

+ f↵

, (↵ = 1, · · · , N)

if u satisfies, for all ⇣ 2 C1

c

(B1

;RN), the following relation.ˆB1

�

Aij

↵�

@j

u�

+Bi

↵�

u�

�

@i

⇣↵

� ⇣↵

�

C i

↵�

@i

u�

+D↵�

u�

�

=

ˆB1

F i

↵

@i

⇣↵

� f↵

⇣↵

,(A.4)

In the two theorems below we collect the regularity and existence results we need

in order to construct the fundamental solution for L.

Theorem A.1. Assume (H1) to (H3). Then the following statements hold.

95

A. THE FUNDAMENTAL SOLUTION OF DIVERGENCE-FORM ELLIPTIC SYSTEMS 96

(1) Suppose 1 < p q < 1 and denote q⇤ = nq/(n + q). Let u 2 W 1,p(B1

;RN) be

a solution to (A.3), with F = (F i

↵

) 2 Lq(B1

;RnN) and f = (f↵

) 2 Lq⇤(B1

;RN).

Then u 2 W 1,q

loc

(B1

;RN) and we have the following estimate.

(A.5) kuk1,q;B1/2

C (kuk1,p;B1 + kfk

q⇤;B1 + kFkq;B1) ,

where C = C(n,N, �, ⇤, µ, p, q).

(2) Suppose u is a solution to (A.3) on B1

\ {0} with F ⌘ 0 and f ⌘ 0. Then for

all 1 < p q < 1, we have u 2 W 1,q

loc

(B1

\ {0}) and(A.6) kuk

1,q;T

0 Ckuk1,p;T

,

where T = Ba

\ Bb

, T 0 = Ba

0 \ Bb

0 and C depends on the same parameters as

in (1) along with the radii 0 < b < b0 < a0 < a < 1.

(3) In (2), if we require that u = 0 on @B1

, then the estimate (A.6) holds true with

0 < b < b0 < 1 and a0 = a = 1.

Theorem A.2. (1) Assume (H1) to (H3) and suppose p, q, q⇤ and F, f are as in

Theorem A.1. Let u 2 W 1,p

0

(B1

;RN) be a solution to (A.3). Then, in fact,

u 2 W 1,q

0

(B1

;RN), and the following estimate holds.

(A.7) kuk1,q;B1 C (kuk

p;B1 + kfkq⇤;B1 + kFk

q;B1) ,

where C depends only on n,N, �, ⇤, µ, p and q.

(2) Suppose that (H4) holds besides (H1) to (H3). Then, given F 2 Lp(B1

;RnN)

and f 2 Lp⇤(B1

;RN) (p⇤ = np/(n + p)), there exists a unique solution u 2W 1,p

0

(B1

;RN) to (A.3), with the following estimate.

(A.8) kuk1,p;B1 C (kfk

p⇤;B1 + kFkp;B1) ,

where C depends on n,N, c0

, �, ⇤, µ and p.

Remark A.3. Theorem A.1 can proved with the help of the results in [Fuc86],

Section 1. Although only the case Bi

↵�

= C i

↵�

= D↵�

⌘ 0 is treated there, the

argument can easily be extended to our setting. As for Theorem A.2, we refer to

[Mor66], Chapter 6.

For later purposes we introduce the adjoint of L, denoted L0 and defined by

(L0v)↵

= @i

�

Aij

↵�

@j

v�

� C i

↵�

v�

�� Bi

↵�

@i

v�

+D↵�

v�

.

We immediately see from the definition that L0 satisfies (H1) to (H4) if and only if L

does. For the remainder of this appendix, we will assume that L (equiva-

lently, L0) satisfies (H1) to (H4).

Under these assumptions, one can construct a fundamental solution for (A.1) in

exactly the same way as done in [Fuc86]. For the reader’s convenience, we briefly

A. THE FUNDAMENTAL SOLUTION OF DIVERGENCE-FORM ELLIPTIC SYSTEMS 97

describe the construction here. By basic functional analysis, we know for each element

l in W�1,p(B1

;RN), the dual of W 1,p

0

0

(B1

;RN) (p0 = p/(p � 1)), we can find a F =

(F i

↵

) 2 Lp(B1

;RnN) such that

(A.9) l(w) =

ˆB1

F i

↵

@i

w↵

, for all w 2 C1

c

(B1

;RN),

Furthermore,

(A.10) klk = inf{kFkp

0;B1 |F satisfies (A.9)}.

This and Theorem A.2(2) imply that, for all l 2 W�1,p(B1

;RN), there is a unique

u 2 W 1,p

0

(B1

;RN) such that L0u = l. Moreover, by (A.8) and (A.10), the operator

T : W�1,p(B1

;RN) ! W 1,p

0

(B1

;RN) taking l to u is bounded. If we choose p > n

and compose T with the embedding W 1,p

0

(B1

;RN) ! C0(B1

;RN), the result is a

bounded operator T : W�1,p(B1

;RN) ! C0(B1

;RN), whose adjoint T 0 is a bounded

operator from M(B1

;RN) =�

C0(B1

;RN)�0

into W 1,p

0

0

(B1

;RN) =�

W�1,p(B1

;RN)�0.

Here M(B1

;RN) denotes the space of RN -valued Radon measure on B1

. We recall

the basic properties of T 0 below.

Proposition A.4. ([Fuc86], Lemma 2.1 and Theorem 3) Let ⌫ 2 M(B1

;RN). Then

(1) T 0(⌫) 2 W 1,p

0

0

(B1

;RN) induces the same element in W 1,1

0

(B1

;RN) regardless of

the choice of p 2 (n,1). We will denote by u⌫

this element, which then lies in

\r<n/(n�1)

W 1,r(B1

;RN).

(2) u⌫

is the unique weak solution to Lu = ⌫, i.e.

(A.11)

ˆB1

�

Aij

↵�

@j

u�

+Bi

↵�

u�

�

@i

⇣↵

� ⇣↵

�

C i

↵�

@i

u�

+D↵�

u�

�

=

ˆB1

⇣↵

d⌫↵

,

for all ⇣ 2 C1

c

(B1

;RN), where (⌫↵

) are the components of ⌫.

(3) For all r 2 [1, n/(n� 1)) there exists C = C(n,N, c0

, �, ⇤, µ, r) such that

(A.12) ku⌫

k1,r;B1 C|⌫|(B

1

).

Definition A.5. We define the fundamental solution G = (G↵�

) : B1

⇥ B1

! RN

2

for (A.1) by G↵�

(·, y) = (u⌫

)�

, with ⌫ = e↵

�y

, where e↵

is the ↵-th basis vector on

RN , and �y

is the Dirac measure supported at y. The fundamental solution (G0↵�

) for

the adjoint operator L0 is defined similarly.

The fundamental solutions G has the following basic properties.

Proposition A.6. (1) For all 1 p < n/(n � 1), we have kG(·, y)k1,p;B1 C,

where C = C(n,N, c0

, �, ⇤, µ, p).

(2) Let y 2 B1/2

and 0 < r 1/4. Then for all 1 q < 1, we have

kG(·, y)k1,q;B1\Br

(y)

C = C(n,N, c0

, �, ⇤, µ, q, r).

A. THE FUNDAMENTAL SOLUTION OF DIVERGENCE-FORM ELLIPTIC SYSTEMS 98

(3) For all x, y 2 B1

with x 6= y and for all 1 ↵, � N , we have G↵�

(x, y) =

G0�↵

(y, x).

(4) Let � denote the diagonal {(x, x)|x 2 B1

}. Then G 2 C0,↵

loc

(B1

⇥ B1

\�;RN

2)

for all 0 < ↵ < 1.

(5) Let ⌫ 2 M(B1

;RN), then G is L1 on B1

⇥B1

with respect to both ⌫ ⇥Hn and

Hn ⇥ ⌫. Moreover, u⌫

(cf. Theorem A.4) can be represented as follows.

u⌫,↵

(y) =

ˆB1

G0↵�

(x, y)d⌫�

(x) =

ˆB1

G�↵

(y, x)d⌫�

(x), for Hn-a.e. y 2 B1

.

Remark A.7. The proofs of the properties listed above can be found in Section 3

of [Fuc86]. Again, although there it is assumed that Bi

↵�

= C i

↵�

= D↵�

⌘ 0, the

argument actually works as long as the operator L satisfies the W 1,p estimates in

Theorems A.1 and A.2.

We now state the main result of this appendix.

Theorem A.8 (see also [Fuc86], Theorem 7). There exists constants C1

, C2

> 0 and

1/8 � R1

> 0, depending only on n,N, c0

, �, ⇤ and µ, such that

(A.13) |G(x, y)| C1

|x� y|2�n + C2

R1+µ�n,

whenever y 2 B1

, R min{R1

, dist(y, @B1

)/4} and x 2 B2R

(y) \ {y}.Corollary A.9. Let C

1

and C2

be as in Theorem A.8. There exists a radius R2

=

R2

(n,N, c0

, �, ⇤, µ) < 1/8, such that

(A.14) |G(x, y)| C1

|x� y|2�n + C2

R1+µ�n

2

,

whenever x, y 2 BR2 and x 6= y.

For the proof of Theorem A.8, we need to introduce two other elliptic operators.

For u : B1

! RN and x0

2 B1/2

, we define�

L(0)u�

↵

= @i

�

Aij

↵�

(x0

)@j

u�

�

,�

L(1)u�

↵

= @i

�

Aij

↵�

@j

u�

�

+ C i

↵�

@i

u�

+ D↵�

u�

,

where D↵�

= D↵�

� K�↵�

and K is a large enough constant depending only on

n,N, �, ⇤ and µ that makes L(1) satisfy (H4). Since L(1) is easily seen to also satisfy

(H1) to (H3), it has a fundamental solution, denoted G(1) and defined as in Definition

A.5, which enjoys the properties listed in Proposition A.6.

On the other hand, it is well-known that the constant-coe�cient operator L(0) has

a fundamental solution E : Rn ! RN

2constructed using the Fourier transform. We

summarize the important properties of E in the two Propositions below. Both results

can be found in Chapter 6 of [Mor66].

A. THE FUNDAMENTAL SOLUTION OF DIVERGENCE-FORM ELLIPTIC SYSTEMS 99

Proposition A.10. (1)�

L(0)E�

↵

= e↵

�0

, where e↵

is the ↵-th coordinate vector

in RN and �0

is the Dirac measure supported at 0 2 Rn.

(2) E is an even function and is positively homogeneous of degree 2�n on Rn\{0}.(3) E is smooth away from the origin and its derivatives satisfy

(A.15) |@⌫E(x)| C|x|2�n�⌫ , C = C(n,N, �, ⇤, |⌫|)where ⌫ = (⌫

1

, · · · , ⌫n

) is a multi-index, @⌫ = @⌫11

· · · @⌫nn

and |⌫| = ⌫1

+ · · ·+⌫n

.

Proposition A.11. Fixing an exponent 1 < p < 1, we have

(1) Given (f↵

) 2 Lp(B1

;RN), define u↵

(x) =´B1

E↵�

(x � y)f�

(y)dy for x 2 B1

.

Then u 2 W 2,p(B1

;RN) and is a weak solution to L(0)u = f . Moreover,

(A.16) kuk2,p;B1 Ckfk

p;B1 , C = C(n,N, �, ⇤, p).

(2) Given (F i

↵

) 2 Lp(B1

;RnN), define v↵

(x) = � ´B1@i

E↵�

(x � y)F i

�

(y)dy for x 2B

1

. Then u 2 W 1,p(B1

;RN) and is a weak solution to L(0)u = @i

F i. Moreover

(A.17) kuk1,p;B1 CkFk

p;B1 , C = C(n,N, �, ⇤, p).

Remark A.12. The homogeneity of E implies that both statements in Proposition

A.11 remain true with B1

replaced by any Br

(x0

). The only changes to be made are

(1) The norms kukk,p;B1 in (A.16) and (A.17) should be replaced by

(A.18) kuk⇤k,p;B

r

(x0)⌘

k

X

i=0

rk�n

p krkukp;B

r

(x0).

(2) The norms kfkp;B1 and kFk

p;B1 should be replaced by r2�n/pkfkp;B

r

(x0) and

r1�n/pkFkp;B

r

(x0), respectively.

Note that the constants C in (A.16) and (A.17) remain the same as before. In

particular they do not dependent on r.

The proof of Theorem A.8 consists of two steps. We first prove Theorem A.8 for

G(1) based on estimates for the fundamental solution of L(0). Then we extend the

result to G using the bounds on G(1) already obtained. We begin with the first step,

where we in fact get bounds for the first-order derivatives of G(1) as well.

Theorem A.13. There exists R0

= R0

(n,N, c0

, �, ⇤, µ) such that Theorem A.8 holds

with G(1) in place of G and R0

in place of R1

. In addition, we have

(A.19)

�

�

�

�

@G(1)

@x(x, y)

�

�

�

�

C1

|x� y|1�n + |x� y|�1R1+µ�n,

whenever y 2 B1

, R min{R0

, dist(y, @B1

)/4} and x 2 BR

(y) \ {y}.

A. THE FUNDAMENTAL SOLUTION OF DIVERGENCE-FORM ELLIPTIC SYSTEMS 100

We first set the stage before going in to the proof. It should be noted that our

arguments is a modification of the proof given in [Fuc86] for the case Bi

↵�

= C i

↵�

=

D↵�

⌘ 0. The basic idea is to obtain the desired estimates on G(1) by writing it as E

plus some perturbation terms which grow no faster than E near the diagonal. To that

end, following [Fuc86], we define the perturbation operator, denoted Tr

, as follows.

(Tr

u)�

(x) =

ˆB

r

(x0)

@i

E�↵

(x� y)�

Aij

↵�

(x0

)� Aij

↵�

(y)�

@j

u�

(y)dy

�ˆB

r

(x0)

E�↵

(x� y)⇣

C i

↵�

(y)@i

u�

(y) + D↵�

(y)u�

(y)⌘

dy

⌘ �T I

r

u�

�

(x)� �T II

r

u�

�

(x),

where x0

2 B1/2

and r < 1/4. From Proposition A.11, we infer that v ⌘ Tr

u is a weak

solution to L(0)v =�

L(0) � L(1)

�

u on Br

(x0

), and that Tr

defines a bounded operator

from W 1,p(Br

(x0

);RN) to itself with respect to the norm k · k⇤1,p;B

r

(x0). Furthermore,

we shall see below that Tr

is a contraction mapping provided r is chosen small enough.

Proposition A.14 (cf. [Fuc86], (1,10) and the remark preceding it). There exists

r0

= r0

(n,N, �, ⇤, µ) such that whenever r r0

, we have

(A.20) kTr

kp

<1

2,

where kTr

kp

denotes the operator norm of Tr

with respect to the k ·k⇤1,p;B

r

(x0)-norm on

W 1,p

0

(B1

;RN).

Proof of Proposition A.14. Suppose r r0

, with r0

to be determined later. We treat

T I

r

and T II

r

separately. The former was already handled in [Fuc86], but we include the

simple argument for completeness. Specifically, by assumption (H2) and Proposition

A.11, for u 2 W 1,p(B1

;RN) we have�

�T I

r

u�

�

⇤1,p;B

r

(x0) Cr1�p/n k(A(x

0

)� A)rukp;B

r

(x0)

C�

oscB

r

(x0) A�

r1�p/nkrukp;B

r

(x0) C⇤rµkuk⇤1,p;B

r

(x0),

where in the last inequality we used assumption (H2).

For T II

r

u, we again apply Proposition A.11 to get�

�T II

r

u�

�

⇤2,p;B

r

(x0) Cr2�p/nkC i

↵�

@i

u�

+ D↵�

u�

kp;B

r

(x0) C⇤rkuk⇤1,p;B

r

(x0).

Next we notice that by the Sobolev inequality and the Holder inequality,

kT II

r

uk⇤1,p;B

r

(x0) CrkT II

r

uk⇤1,p

⇤;B

r

(x0) CrkT II

r

uk⇤2,p;B

r

(x0).

A. THE FUNDAMENTAL SOLUTION OF DIVERGENCE-FORM ELLIPTIC SYSTEMS 101

Therefore we arrive at kT II

r

uk⇤1,p;B

r

(x0) C⇤r2kuk⇤

1,p;B

r

(x0). Combining the estimates

for T I

r

u and T II

r

u, we conclude that

(A.21) kTr

uk⇤1,p;B

r

(x0) C⇤rµkuk⇤

1,p;B

r

(x0),

for all u 2 W 1,p(B1

;RN). Choosing r0

small enough completes the proof. ⇤

We now proceed to prove Theorem A.13. The proof will be given in two parts:

First we establish the zeroth-order estimate (A.13) for G(1), then we prove the first-

order derivative estimate (A.19).

Proof that (A.13) holds for G(1). Throughout the proof, we will denote by C any con-

stant that depends only on n,N, c0

, �, ⇤ and µ, and specify other types of dependence

when necessary.

As in [Fuc86], we fix p = n/(n� µ) and q = n/(1� µ). Note that p < n/(n� 1)

whereas q > n. Next, let y 2 B1/2

and r min{r0

/2, 1/8}, with r0

given by Proposi-

tion A.14. In order to simplify notations, we assume with out loss of generality that

y = 0. Furthermore, for a fixed index � we write write G(1)

�

(·) for (G�↵

(·, 0))1↵N

.

Similarly, we write E�

(·) for (E�↵

(·))1↵N

, where E is the fundamental solution of

L(0) with x0

= 0.

We then observe that both G(1)

�

� T2r

G(1)

�

and E�

are weak solutions on B2r

to�

L(0)u�

= e�

�0

. For E�

, this follows by definition, whereas for G(1)

�

� T2r

G(1)

�

we have

L0G(1)

�

� L(0)

�

T2r

G(1)

�

�

= L0G(1)

�

� �L(0) � L(1)

�

G(1)

�

= L(1)G1

�

= e�

�0

.

Hence there exists w 2 W 1,p(B2r

;RN), with L(0)w = 0 in B2r

, such that

(A.22) G(1)

�

= E�

+ T2r

G(1)

�

+ w.

Since r r0

/2, we have by Proposition A.14 that kT2r

kp

< 1/2 and thus (I � T2r

)

has a bounded inverse on W 1,p(B2r

;RN) given byP1

l=0

T l

2r

. Applying this to (A.22),

we see that

(A.23) G(1)

�

= E�

+1X

l=1

T l

2r

E�

+1X

l=0

T l

2r

w,

as functions in W 1,p(B2r

;RN). Among the terms on the right-hand side, the first term

already has the desired growth by (A.15). Therefore the proof boils down to showing

that the remaining two terms grows no faster than |x|2�n near the origin.

Lemma A.15 (cf. [Fuc86], Lemma 4.1). We have w 2 W 1,q(B2r

;RN) and kwk⇤1,q;B2r

Cr1�n/p.

A. THE FUNDAMENTAL SOLUTION OF DIVERGENCE-FORM ELLIPTIC SYSTEMS 102

Proof of Lemma A.15. We will estimate kwk⇤1,q;B

r

and kwk⇤1,q;B2r\Br

separately. Since

L(0)w = 0 on B2r

, a scaled version of Theorem A.1(2) gives

(A.24) kwk⇤1,q;B

r

Ckwk⇤1,p;B2r

.

To estimate kwk⇤1,p;B2r

, note that by the triangle inequality,

kwk⇤1,p;B2r

kG(1)

�

k⇤1,p;B2r

+ kT2r

G(1)

�

k⇤1,p;B2r

+ kE�

k⇤1,p;B2r

.

The terms kE�

k⇤1,p;B2r

and kG(1)

�

k⇤1,p;B2r

are estimated exactly as in [Fuc86]. To be

precise, by (A.15) and a direct computation we obtain kEkp;B2r Cr2�n+n/p and that

krEkp;B2r Cr1�n+n/p. Consequently,

(A.25) kEk⇤1,p;B2r

= r�n/pkEkp;B2r + r1�n/pkrEk

p;B2r Cr2�n Cr1�n/p,

where in the last inequality we used the fact that p < n/(n� 1). On the other hand,

with the help of Proposition A.6(1) and the Sobolev inequality, we get

kG(1)k⇤1,p;B2r

= r�n/pkG(1)kp;B2r + r1�n/pkrG(1)k

p;B2r(A.26)

r1�n/pkG(1)kp

⇤;B2r + r1�n/pkrG(1)k

p;B1

r1�n/p

�kG(1)kp

⇤;B1 + krG(1)k

p;B1

�

Cr1�n/pkrG(1)kp;B1 Cr1�n/p.

It follows from (A.26) and Proposition A.14 that kT2r

G(1)

�

k⇤1,p;B2r

Cr1�n/p as well.

Putting these back into (A.24), we obtain

(A.27) kwk⇤1,q;B

r

Cr1�n/p.

It remains to show that kwk1,q;B2r\Br

Cr1�n/p. Again by the triangle inequality,

(A.28) kwk⇤1,q;B2r\Br

kG(1)

�

k⇤1,q;B2r\Br

+ kT2r

G(1)

�

k⇤1,q;B2r\Br

+ kE�

k⇤1,q;B2r\Br

.

Next we use Theorem A.1(2) and (A.26) (with 2r replaced by 3r) to obtain

(A.29) kG(1)k⇤1,q;B2r\Br

CkG(1)k⇤1,p;B3r\B

r/2 CkG(1)k⇤

1,p;B3r Cr1�n/p.

On the other hand, by the estimate (A.15) and a computation similar to the one for

(A.25), we get

(A.30) kEk⇤1,q;B2r\Br

Cr1�n/p.

As for the term kT2r

G(1)

�

k⇤1,q;B2r\Br

, by its definition we can write

kT2r

G(1)

�

k⇤1,q;B2r\Br

kT I

2r

G(1)

�

k⇤1,q;B2r\Br

+ kT II

2r

G(1)

�

k⇤1,q;B2r\Br

.

The first term on the right-hand side was already handled in [Fuc86], and the con-

clusion is that

(A.31) kT I

2r

G(1)

�

k⇤1,q;B2r\Br

Cr1�n/p.

A. THE FUNDAMENTAL SOLUTION OF DIVERGENCE-FORM ELLIPTIC SYSTEMS 103

For the second term, we break the definition of T II

2r

u into two integrals, one over Br/2

and the other over B2r

\Br/2

. That is, we write

�

T II

2r

G(1)

�

�

⌫

(x) =

ˆB

r/2

E⌫↵

(x� y)⇣

C i

↵�

(y)@i

G(1)

��

(y) + D↵�

(y)G(1)

��

(y)⌘

dy

+

ˆB2r\B

r/2

E⌫↵

(x� y)⇣

C i

↵�

(y)@i

G(1)

��

(y) + D↵�

(y)G(1)

��

(y)⌘

dy

⌘ 1

(x) + 2

(x).

To estimate 1

, notice that when x 2 B2r

\Br

and y 2 Br/2

, E(x�y) is smooth in both

variables and we have by (A.15) that |E(x� y)| Cr2�n and |rx

E(x� y)| Cr1�n.

Thus we have

|r 1

(x)| Cr1�n⇤kG(1)k1,1;B

r/2 Cr1�n/pkG(1)k

1,p;B

r/2 Cr1�n/p,

where in the last inequality we again used Proposition A.6(1). Integrating the above

pointwise estimate, we obtain

kr 1

kq;B2r\Br

Cr1�n/prn/q = Cr2�n Cr1�n/p.

A similar computation yields k 1

kq;B2r\Br

Cr2�n/p, and we get

(A.32) k 1

k⇤1,q;B2r\Br

Cr1�n/qr1�n/p Cr1�n/p,

where the last inequality follows from the fact that q > n. As for 2

, observe that by

Proposition A.11(1) and the remark after it,

(A.33) k 2

k⇤1,q;B2r

CkG(1)

�

k⇤1,q;B2r\Br

Cr1�n/p,

where we used the estimate (A.29) in the last inequality. Putting together (A.32) and

(A.33), we conclude that

(A.34) kT II

2r

G(1)

�

k⇤1,q;B2r\Br

Cr1�n/p.

Substituting the estimates (A.29), (A.30), (A.31) and (A.34) back into (A.28), we get

(A.35) kwk⇤1,q;B2r\Br

Cr1�n/p.

The proof of Lemma A.15 is now complete in view of (A.27) and (A.35). ⇤

It follows from Lemma A.15 and Proposition A.14 that1P

l=0

T l

2r

w 2 W 1,q(B2r

;RN)

and

(A.36)

�

�

�

�

�

1X

l=0

T l

2r

w

�

�

�

�

�

⇤

1,q;B2r

X

2�lkwk⇤1,q;B2r

Cr1�n/p.

A. THE FUNDAMENTAL SOLUTION OF DIVERGENCE-FORM ELLIPTIC SYSTEMS 104

Since q > n, by Sobolev embedding we have

(A.37)

�

�

�

�

�

1X

l=0

T l

2r

w

�

�

�

�

�

0;B2r

C

�

�

�

�

�

1X

l=0

T l

2r

w

�

�

�

�

�

⇤

1,q;B2r

Cr1�n/p = Cr1+µ�n.

Note that this accounts for the second term on the right-hand side of (A.13).

Next we turn to estimating the term1P

l=1

T l

2r

E�

. Borrowing the notation in [Fuc86],

we write ul

for T l

2r

E�

, for l = 0, 1, 2, · · · .Lemma A.16 (cf. [Fuc86], Lemma 4.3). The following estimates hold for all l =

0, 1, 2, · · · .(i) |u

l

(x)| C(Crµ)l|x|2�n, for all x 2 B2r

\ {0}.(ii) |ru

l

(x)| C(Crµ)l|x|1�n, for all x 2 B2r

\ {0}.(iii) |ru

l

(x) � rul

(y)| C(Crµ)l|x � y|µ max(|x|1�n�µ, |y|1�n�µ), for all x 6= y 2B

2r

\ {0}.Proof of Lemma A.16. Unsurprisingly, the proof proceeds by induction on l. For

l = 0, ul

is just E�

, and we have the desired estimates from (A.15). Now, assume

that the claim holds for l � 1. Then by definition we can write ul

as

(ul

)⌫

(x) =

ˆB2r

@i

E⌫↵

(x� y)�

Aij

↵�

(0)� Aij

↵�

(y)�

@j

(ul�1

)�

(y)dy

�ˆB2r

E⌫↵

(x� y)⇣

C i

↵�

(y)@i

(ul�1

)�

(y) + D↵�

(y) (ul�1

)�

(y)⌘

dy

⌘ (ul,1

)⌫

(x)� (ul,2

)⌫

(x).

The potential estimates in [Fuc86] carries over word for word to our setting to show

that ul,1

satisfies (ii) and (iii) under the induction hypothesis. We thus proceed to

prove (i) for ul

. Below, to simplify notations, we will often drop the Greek and Latin

indices when writing convolution integrals. Now observe that

ul,1

(x) =

ˆB2r

rE(x� y) (A(0)� A(y))rul�1

(y)dy.

Taking absolute values, recalling (A.15) and using the induction hypothesis (ii) for

ul�1

, we obtain

|ul,1

(x)| C

ˆB2r

|x� y|1�n[A]µ,B1r

µ|rul�1

(y)|dy(A.38)

C(Crµ)l�1⇤rµˆB2r

|x� y|1�n|y|1�ndy.

A. THE FUNDAMENTAL SOLUTION OF DIVERGENCE-FORM ELLIPTIC SYSTEMS 105

The evaluate the last integral, we need the following fact: For 0 < ⌧, � < n, there

exists a constant C = C(n, �, ⌧) such that for x 2 B1

, we have

(A.39)

ˆB1

|x� y|��n|y|⌧�ndy

8

>

<

>

:

C|x|�+⌧�n , � + ⌧ < n.

C(1 + log |x|) , � + ⌧ = n.

C , � + ⌧ > n.

For the simple proof of this fact, we refer the reader to [Aub82]. Returning to the

main line of reasoning, we see from (A.39) and (A.38) that

(A.40) |ul,1

(x)| C⇤rµ(Crµ)l�1C|x|2�n C(Crµ)l|x|2�n.

Therefore, (i) holds for ul,1

. Next, we note that a similar reasoning applied to the

following integral and the defining integral of ul,2

yields (i) and (ii) for ul,2

.

(A.41) rul,2

(x) =

ˆB2r

rE(x� y)⇣

C(y)rul�1

(y) + D(y)ul�1

(y)⌘

dy.

Finally, to get (iii) for ul,2

, we note that it su�ces to consider x and z satisfying

(A.42) ⇢ ⌘ |x� z| 1

8min{|x|, |z|},

for if the reverse inequality holds, then (iii) follows easily from (ii). Moreover, we will

assume that |x| |z| and write ⇠ = (x+ z)/2. Note that (A.42) then implies that

(A.43)15

16|x| |x|� |x� z|/2 |⇠| |x|+ |x� z|/2 17

16|x|.

We will repeatedly use this fact in the arguments to follow.

We now breakrul,2

(x)�rul,2

(z) into four integrals and estimate them separately.

More precisely,

rul,2

(x)�rul,2

(z) =

ˆB

⇢

(⇠)

rE(x� y)F (y)dy �ˆB

⇢

(⇠)

rE(z � y)F (y)dy

+

ˆB|⇠|/2

(rE(x� y)�rE(z � y))F (y)dy

+

ˆB2r\(B⇢

(⇠)[B|⇠|/2)

(rE(x� y)�rE(z � y))F (y)dy

⌘ I � II + III + IV,

where we let F (y) = C(y)rul�1

(y) + D(y)ul�1

(y) to save space. Note that by the

induction hypotheses, |F (y)| C(Crµ)l�1⇤(1 + |y|)|y|1�n C(Crµ)l�1|y|1�n.

Integrals (I) and (II) are estimated in identical ways, so we only give the arguments

for (I). Taking absolute values, we see that

|(I)| C(Crµ)l�1

ˆB

⇢

(⇠)

|x� y|1�n|y|1�ndy(A.44)

A. THE FUNDAMENTAL SOLUTION OF DIVERGENCE-FORM ELLIPTIC SYSTEMS 106

C(Crµ)l�1

ˆB3⇢(x)

|y � x|1�n|x|1�ndy

C(Crµ)l�1|x|1�n⇢ C(Crµ)l�1rµ|x|1�n�µ⇢µ

C(Crµ)l|x|1�n�µ⇢µ.

In the second inequality above, we used the fact that

|y| � |⇠|� ⇢ � |x|� 2⇢ � 3

4|x|, whenever y 2 B

⇢

(⇠) and (A.42) holds.

Also, in the second to last inequality, we wrote |x|1�n = |x|1�n�µ|x|µ and estimated

the second factor by rµ.

Next we estimate (III). First notice that by the mean-value theorem, there exists

x0 2 B⇢

(x) such that

(A.45) |rE(x� y)�rE(z � y)| ⇢�

�r2E(x0 � y)�

� C⇢|x0 � y|�n.

Now observe that for all y 2 B|⇠|/2 and x0 2 B⇢

(x), from (A.43) we have

(A.46) |x0 � y| � |x|� ⇢� |⇠|/2 � 1

4|x|.

Therefore we obtain |rE(x� y)�rE(z � y)| C⇢|x|�n. Plugging this into the

defining integral of (III), we have

|(II)| C⇢

ˆB|⇠|/2

|x|�n|F (y)|dy C⇢

ˆB|⇠|/2

|x|�nC(Crµ)l�1|y|1�ndy

C(Crµ)l�1⇢|x|�n|⇠| C(Crµ)l�1⇢µrµ|x|�n�µ|x| C(Crµ)l⇢µ|x|1�n�µ,

where in the second to last inequality we used (A.43) and that |x|�n |x|�n�µrµ.

Finally we estimate (IV). Again we use the mean-value theorem to find x0 2B⇢/2

(⇠) such that (A.45) holds. To estimate the right-hand side in (A.45) in a useful

way, we notice that for y 2 B2r

\ (B⇢

(⇠)[B|⇠|/2) and x0 2 B⇢/2

(⇠), we have |x0 � ⇠| ⇢/2 |y � ⇠|/2 and hence

|y � x0| � |y � ⇠|� |x0 � ⇠| � 1

2|y � ⇠|.

Therefore |x0 � y|�n 2n|y � ⇠|�n, and (IV) can be estimated as follows.

(IV ) C⇢

ˆB2r\(B⇢

(⇠)[B|⇠|/2)

|⇠ � y|�n|F (y)|dy

C⇢

ˆB2r\(B⇢

(⇠)[B|⇠|/2)

|⇠ � y|�nC(Crµ)l�1|y|1�ndy

C⇢(Crµ)l�1|⇠|1�n

ˆB2r\(B⇢

(⇠)[B|⇠|/2)

r1�µ|⇠ � y|�n�1+µdy

A. THE FUNDAMENTAL SOLUTION OF DIVERGENCE-FORM ELLIPTIC SYSTEMS 107

C⇢(Crµ)l�1|x|1�nr1�µ⇢µ�1

C⇢µ(Crµ)l�1|x|1�n�µr C⇢µ(Crµ)l|x|1�n�µ.

In the third inequality above, we used the fact that |y| � |⇠|/2 on the domain of

integration, and that

(A.47) |⇠ � y|�n = |⇠ � y|�n�1+µ|⇠ � y|1�µ r1�µ|⇠ � y|�n�1+µ.

Having estimated (I), (II), (III) and (IV), we conclude that (iii) holds for ul,2

, thus

finishing the induction step. The proof of Lemma A.16 is now complete. ⇤

We now conclude the proof of (A.13) for G(1) as follows. Requiring that r R0

⌘min{r

0

/2, 1/8, (2C)�1/µ}, by the estimate (i) in Lemma A.16, we have

(A.48)1X

l=1

T l

2r

E�

(x) C1X

l=1

2�l|x|2�n C|x|2�n.

Combining this with (A.37) and recalling (A.23), we arrive at

(A.49) |G(1)

�

(x)| |E�

(x)|+ C|x|2�n + Cr1+µ�n C|x|2�n + Cr1+µ�n.

The proof of (A.13) for G(1) is now complete. ⇤

Proof of the first-order derivative estimate (A.19). Our proof is again derived from

that in [Fuc86]. In short, having proven (A.13), we will show that (A.19) follows

from standard C1,µ-estimates for elliptic systems and a scaling argument. As in the

previous proof, we assume without loss of generality that y = 0, and we write G(1)

�

(·)for (G

�↵

(·, 0))1↵N

.

Fix r R0

and choose a point z 2 Br

. We let ⇢ = |z|/2 and rescale G(1)

�

by

letting G(x) ⌘ G(1)

�

(z + ⇢x). Since G(1)

�

satisfies L(1)G(1)

�

= 0 on B⇢

(z), we infer that

LG = 0 on B1

, where the operator L has the form

(A.50)⇣

Lu⌘

↵

= @i

⇣

Aij

↵�

@ju�

⌘

+ C i

↵�

@i

u�

+ D↵�

u�

,

with coe�cients given by

(A.51)

8

>

<

>

:

Aij

↵�

(x) = Aij

↵�

(z + ⇢x)

C i

↵�

(x) = ⇢C i

↵�

(z + ⇢x)

D↵�

(x) = ⇢2D↵�

(z + ⇢x).

We see immediately from the definition that L also satisfies (H1) to (H3) with the same

parameters �, ⇤ and µ. Since LG = 0 on B1

, Schauder theory for divergence-form

elliptic systems (see [Mor66] or [GM12]) implies that G 2 C1,µ

loc

(B1

;RN). Further-

more, the following estimate holds.

(A.52) |rG|0;B1/2

CkGk1,2;B3/4

.

A. THE FUNDAMENTAL SOLUTION OF DIVERGENCE-FORM ELLIPTIC SYSTEMS 108

Note that the right-hand side can be replaced by kGk2;B1 since the Caccioppoli in-

equality implies that

(A.53) krGk2;B3/4

CkG� (G)0,1

k2;B1 CkGk

2;B1 .

Scaling back and recalling the definition of ⇢, we obtain

(A.54) |z||rG(1)

�

|0;B|z|/4(z) C|z|�n/2kG(1)

�

k2;B|z|/2(z).

Since |z| < r, it follows that B|z|/2(z) ⇢ B2r

\{0}. Thus we may use the bound (A.13)

for G(1)

�

to estimate the right-hand side of (A.54) by

C|z|�n/2

�

C|z|2�n + r1+µ�n

� |z|n/2 = C�|z|2�n + r1+µ�n

�

.

Plugging this back into (A.54) and dividing both sides by |z|, we arrive at

(A.55) |rG(1)

�

|0;B|z|/4(z) C

�|z|1�n + |z|�1r1+µ�n

�

.

In particular, (A.19) holds, and we are done. ⇤

Before proceeding with the proof of Theorem A.8, let us present some further

properties of G(1) that we will need.

Proposition A.17. (1) G(1) : B1

⇥ B1

! RN

2is locally C1 on B

1

⇥ B1

\ �.

Moreover, for x 6= y, we have

(A.56) @(1)i

G(1)

↵�

(x, y) = @(2)i

G0(1)�↵

(y, x),

where @(1)i

(resp., @(2)i

) denotes partial di↵erentiation with respect to i-th variable

in the first slot (resp., second slot), and G0(1) denotes the fundamental solution

for L0(1), the adjoint of L(1).

(2) For p 2 [1,1) and F = (F i

↵

) 2 Lp(B1

;RnN), let u 2 W 1,p

0

(B1

;RN) be the

unique solution to L(1)u = @i

F i. Then, for Hn-a.e. y 2 B1

, we have

(A.57) u�

(y) =

ˆB1

@(2)i

G(1)

↵�

(y, x)F i

↵

(x)dx.

(3) For p 2 [1,1) and f = (f↵

) 2 Lp⇤(B1

;RN), let u 2 W 1,p

0

(B1

;RN) be the unique

solution to L(1)u = f . Then, for Hn-a.e. x 2 B1

, we have

(A.58) u�

(x) =

ˆB1

G(1)

↵�

(x, y)f↵

(y)dy.

Proof of Proposition A.17(1). For (1), we start by noting that (A.56) is an immediate

consequence of Proposition A.6(3). Next, to see that G1 2 C1

loc

(B1

⇥ B1

\ �;RN

2),

we will argue that for each ⇢ > 0, G(1) is C1 on the set

A⇢

⌘ {(x, y) 2 B1

⇥ B1

| |x� y| > ⇢}.

A. THE FUNDAMENTAL SOLUTION OF DIVERGENCE-FORM ELLIPTIC SYSTEMS 109

To that end, let us fix (x0

, y0

) 2 A⇢

, and look at the family of functions�

G(1)(·, y) y2B

⇢/2(y0).

Below, for convenience, we denote the restriction of G(1)(·, y) to B⇢/2

(x0

) by uy

.

Claim. The family {uy

}y2B

⇢/2(y0)is bounded in the | · |

1,µ;B

⇢/3(x0)-norm.

Proof of the claim. Notice that since |x0

� y0

| > ⇢, for all y 2 B⇢/2

(y0

), the functions

uy

actually solve L(1)uy

= 0 on B⇢/2

(x0

). By the interior C1,µ-estimates for divergence-

form elliptic systems, we infer that

(A.59) |uy

|1,µ;B

⇢/3(x0) Ckuy

k1,2;B

⇢/2(x0).

Since |x0

� y0

| > ⇢, we see from Proposition A.6 (2) with q = 2 that

kuy

k1,2;B

⇢/2(x0) kuy

k1,2;B1\B

⇢/2(y0) C

⇢

.

The claim is proved upon combining the above inequalities. ⇤

From the claim and the Arzela-Ascoli theorem, we see that for each sequence yi

!y0

, a subsequence of uy

i

converges in C1(B⇢/3

(x);RN

2) to a limit in C1,µ(B

⇢

(3);RN

2).

By Proposition A.6(4), this limit must be uy0 . Since there is a unique subsequential

limit, we conclude that uy

converges to uy0 in C1-topology as y ! y

0

.

Now let (xi

, yi

) ! (x0

, y0

) as i ! 1. Then for i su�ciently large, we have

|@(1)i

G(1)(xi

, yi

)� @(1)i

G(1)(x0

, y0

)| |@(1)i

G(1)(xi

, yi

)� @(1)i

G(1)(xi

, y0

)|+ |@(1)

i

G(1)(xi

, y0

)� @(1)i

G(1)(x0

, y0

)| |u

y

i

� uy0 |1,0;B

⇢/3(x0) + |xi

� x0

|µ[uy0 ]1,µ;B

⇢/3(x0)

! 0 as i ! 1.

Thus we’ve shown that @(1)i

G(1) is continuous on A⇢

. Applying the same argument to

G0(1) and using the relation (A.56), we see that the same conclusion holds for @(2)i

G(1).

In view of this and Proposition A.6(4), we conclude that G(1) is C1 on A⇢

for any

⇢ > 0. The proof of Proposition A.17(1) is now complete. ⇤

Next we turn to assertions (2) and (3) of Proposition A.17. For this purpose we

need to introduce the mollified fundamental solutions, denoted G(1)⇢(·, y), defined for

⇢ > 0 to be unique solution in \1r<n/(n�1)

W 1,r(B1

;RN) to

(A.60) L(1)u =1

|B⇢

|�B

⇢

(y)

dHn.

The mollified version of the adjoint fundamental solution G0(1) is defined similarly. In

what follows, we will frequently use the following property of G(1)⇢.

A. THE FUNDAMENTAL SOLUTION OF DIVERGENCE-FORM ELLIPTIC SYSTEMS 110

Proposition A.18 ([Fuc86], Corollary to Theorem 4). For x, y 2 B1

and 0 < ⇢ <

dist(y, @B1

), the following holds.

(A.61) G(1)⇢

↵�

(x, y) =

B

⇢

(y)

G0(1)�↵

(z, x)dz =

B

⇢

(y)

G(1)

↵�

(x, z)dz,

where the�B

⇢

(y)

denotes the average over B⇢

(y).

Remark A.19. The proofs given in [Fuc86] were based solely on the W 1,p-estimates

in Theorems A.1 and A.2, and therefore apply to our setting.

Proof of Proposition A.17 (2). To simplify notations, we fix an index � and denote

G(1)⇢

�

(·, y) (resp., G(1)

�

(·, y)) by u⇢y

(resp., uy

), and G0(1)⇢�

(·, y) (resp., G01�

(·, y)) by v⇢y

(resp., vy

).

Now, let u be as in the hypothesis of (2), and use v⇢y

as a test function in the

equation L(1)u = @i

F i. Then we getˆB1

Aij

↵�

@j

u�

@i

v⇢y,↵

� v⇢y,↵

(C i

↵�

@i

u�

+ D↵�

u�

) =

ˆB1

F i

↵

@i

v⇢y,↵

.

Note that the left-hand side can understood as testing the system L0(1)v⇢y

= |B⇢

|�1�B

⇢

(y)

dHn

against u. Thus we getˆB1

Aij

↵�

@j

u�

@i

v⇢y,↵

� v⇢y,↵

(C i

↵�

@i

u�

+ D↵�

u�

) =

B

⇢

(y)

u�

.

Putting the two above equations together, we obtain

(A.62)

B

⇢

(y)

u�

(z)dz =

ˆB1

F i

↵

(x)@i

v⇢y,↵

(x)dx.

Claim. @i

v⇢y,↵

(x) =�B

⇢

(y)

@(2)i

G(1)

↵�

(z, x)dz, for x 2 B1

.

Proof of the claim. The idea of proof is borrowed from [GT83]. Let � : [0,1) ![0, 1] be a cut-o↵ function with �(t) = 1 for t 1 and �(t) = 0 for t � 2 and let

(A.63) J✏

(x) =

B

⇢

(y)

(1� �(✏�1|x� z|))G(1)

↵�

(z, x)dz.

(A.64) J(x) =

B

⇢

(y)

G(1)

↵�

(z, x)dz; Ji

(x) =

B

⇢

(y)

@(2)i

G(1)

↵�

(z, x)dz.

We first note that, for ✏ < min{R0

, dist(x, @B1

)/4}, where R0

is given by Theorem

A.13, there holds

|J(x)� J✏

(x)| B

⇢

(y)

�(✏�1|z � x|)|G(1)

↵�

(z, x)|dz

C|B⇢

|�1

ˆB2✏(x)

|x� z|2�n +R1+µ�n

0

dz

A. THE FUNDAMENTAL SOLUTION OF DIVERGENCE-FORM ELLIPTIC SYSTEMS 111

C⇢

✏2 ! 0, as ✏! 0.

Thus J✏

converges uniformly locally on B1

to J as ✏ goes to zero. Next, since G(1)

↵�

is

C1 away from the diagonal by Proposition A.17(1), we have

@i

J✏

(x) =

B

⇢

(y)

(1� �(✏�1|z � x|))@(2)i

G(1)

↵�

(z, x)dz

� B

⇢

(y)

@i

�

�(✏�1|z � x|)�G(1)

↵�

(z, x)dz

⌘ I � II.

For (II), we observe that, again by Theorem A.13, for small enough ✏,

|(II)| C⇢

✏�1

ˆB2✏(x)\B✏

(x)

|x� z|2�n +R1+µ�n

0

dz C⇢

✏,

which tends to zero uniformly in x as ✏ goes to zero. On the other hand, by (A.19)

we estimate

|I � J(x)| = B

⇢

(y)

�(✏�1|z � x|)|@(2)i

G(1)

↵�

(z, x)|dz

C⇢

ˆB2✏(x)\B✏

(x)

|x� z|1�n + |x� z|�1R1+µ�n

0

dz

C⇢

✏! 0 uniformly in x as ✏! 0.

Combining the two estimates above, we infer that @i

J✏

converges uniformly to Ji

as ✏

goes to zero. Since J✏

and @i

J✏

converge uniformly in x to J and Ji

, respectively, we

conclude that J is di↵erentiable in x, and that @i

J(x) = Ji

(x). The claim is proved

upon noticing that J(x) = v⇢y

(x) by Proposition A.18 applied to G0(1)⇢. ⇤

Going back to (A.62), we see from the claim that the right-hand side equals

(A.65)

ˆB1

B

⇢

(y)

F i

↵

(x)@(2)i

G(1)

↵�

(z, x)dz

!

dx.

By (A.56) with G0(1) in place of G(1) and using Proposition A.6(1), we see that

k@(2)i

G(1)(·, x)k1;B1 is bounded independent of x. Since F 2 Lp, we then have

(A.66)

ˆB1

B

⇢

(y)

|F i

↵

(x)|�

�

�

@(2)i

G(1)

↵�

(z, x)�

�

�

dz

!

dx < 1.

Therefore we may switch the order of integration in (A.65) to getˆB1

B

⇢

(y)

F i

↵

(x)@(2)i

G(1)

↵�

(z, x)dz

!

dx =

B

⇢

(y)

✓ˆB1

F i

↵

(x)@(2)i

G(1)

↵�

(z, x)dx

◆

dz.

By the finiteness condition (A.66), the inner-integral on the right-hand side is an L1-

function in z, and therefore, using the Lebesgue di↵erentiation theorem and recalling

A. THE FUNDAMENTAL SOLUTION OF DIVERGENCE-FORM ELLIPTIC SYSTEMS 112

(A.62), for Hn-a.e. y we have

(A.67) lim⇢!0

B

⇢

(y)

u�

(z)dz =

ˆB1

F i

↵

(x)@(2)i

G(1)

↵�

(y, x)dx.

However, since u 2 Lp(B1

;RN), the left-hand side of the above identity equals u�

(y)

for Hn-a.e. y (again by Lebesgue’s di↵erentiation theorem). Hence, for Hn-a.e. y,

u�

(y) =

ˆB1

F i

↵

(x)@(2)i

G(1)

↵�

(y, x)dx.

⇤

Proof of Proposition A.17 (3). We will use the same notations as in the proof of

Proposition A.17 (2). Let u be as in the hypothesis of (3) and use v⇢y

as a test

function in the system L(1)u = f . Then we getˆB1

Aij

↵�

@j

u�

@i

v⇢y,↵

� v⇢y,↵

(C i

↵�

@i

u�

+ D↵�

u�

) =

ˆB1

f↵

v⇢y,↵

.

Again, we can view the left-hand side as testing the system L0(1)v⇢y

= |B⇢

|�1�B

⇢

(y)

dHn

against u. Thus we infer that

(A.68)

B

⇢

(y)

u�

(z)dz =

ˆB1

f↵

(x)v⇢y,↵

(x)dx.

By (A.61) with G0(1) in place of G(1), we find that

(A.69)

ˆB1

f↵

(x)v⇢y,↵

(x)dx =

ˆB1

B

⇢

(y)

f↵

(x)G(1)

↵�

(z, x)dz

!

dx.

As in the previous proof, since f 2 Lp and kG(1)

↵�

(·, x)k1;B1 is bounded uniformly in x

by Proposition A.6 (1) and (3), the order of integration on the right-hand side can

be switched, resulting in

(A.70)

ˆB1

f↵

(x)v⇢y,↵

(x)dx =

B

⇢

(y)

✓ˆB1

f↵

(x)G(1)

↵�

(z, x)dx

◆

dz.

Combining this wit (A.68), letting ⇢ go to zero and using the Lebesgue di↵erentiation

theorem, we see that for Hn-a.e. y 2 B1

,

(A.71) u�

(y) =

ˆB1

f↵

(x)G(1)

↵�

(y, x)dx.

⇤

Remark A.20. The same proof as the one given above shows that Proposition A.17

holds with L in place of L(1) and G in place of G(1).

At this point we are almost ready to prove Theorem A.8. The proof will be

another perturbation argument, but this time based on estimates for G(1) instead of

A. THE FUNDAMENTAL SOLUTION OF DIVERGENCE-FORM ELLIPTIC SYSTEMS 113

E. As in the proof of Theorem A.13, we need to introduce a perturbation operator

and show that locally it is a contraction mapping.

Proposition A.21. Take a point x0

2 B1/2

and a radius r 2 (0, 1/4). For a function

u : Br

(x0

) ! RN and for x 2 Br

(x0

) define

(Tr

u)�

(x) = �ˆB

r

(x0)

@(2)i

G(1)

↵�

(x, y)Bi

↵�

(y)u�

(y)dy(A.72)

�K

ˆB

r

(x0)

G(1)

↵�

(x, y)u↵

(y)dy.

Then the following are true.

(1) If u 2 W 1,p(Br

(x0

);RN), then Tr

u is a weak solution in W 1,p(Br

(x0

);RN) to

(A.73)�

L(1)u�

↵

= �@i

�

Bi

↵�

u�

��Ku↵

.

(2) Write kTr

kp

for the operator norm of Tr

: W 1,p(Br

(x0

);RN) ! W 1,p(Br

(x0

);RN)

with respect to the norm k ·k⇤1,p;B

r

(x0). Then there exists r

1

= r1

(n,N, c0

, �, ⇤, µ)

such that whenever r r1

, we have kTr

kp

< 1/2.

Proof. Assertion (1) follows straight from (2) and (3) of Proposition A.17. Thus we

will focus on proving assertion (2). To that end, notice that the integrals in (A.72) in

fact make sense for x 2 B1

, and Proposition A.17 tells us that the resulting function

on B1

, still denoted Tr

u, is the weak solution in W 1,p

0

(B1

;RN) to (A.73), with u

extended to be zero outside of Br

(x0

). Thus, the global estimates in Theorem A.2

implies that

krTr

ukp;B1 C

�

⇤kukp;B

r

(x0) +Kkukp⇤;Br

(x0)

�

(A.74)

C(⇤+Kr)kukp;B

r

(x0).

Thus, absorbing K into the constant C and using the fact that kukp;B

r

(x0) kukp;B1 ,

we have

(A.75) r1�n/pkrTr

ukp;B

r

(x0) Cr1�n/pkrukp;B1 Crr�n/pkuk

p;B

r

(x0).

Moreover, since Tr

u 2 W 1,p

0

(B1

;RN), by the Sobolev inequality we have kTr

ukp

⇤;B1

CkrTr

ukp;B1 , and thus

r�n/pkTr

ukp;B

r

(x0) r1�n/pkTr

ukp

⇤;B

r

(x0) r1�n/pkTr

ukp

⇤;B1(A.76)

Cr1�n/pkrTr

ukp;B1 Crr�n/pkuk

p;B

r

(x0).

From the above inequalities and recalling the definition of k · k⇤1,p;B

r

(x0), we arrive at

(A.77) kTr

uk⇤1,p;B

r

(x0) Crkuk⇤

1,p;B

r

(x0).

Taking r1

= (2C)�1 completes the proof. ⇤

A. THE FUNDAMENTAL SOLUTION OF DIVERGENCE-FORM ELLIPTIC SYSTEMS 114

Proof of Theorem A.8. Without loss of generality, we assume that y = 0. More-

over, we adopt the same notational simplifications as in the proof of Theorem A.13.

Hence, for a fixed index �, we will write G�

(·) and G(1)

�

(·) for (G�↵

(·, 0))1↵N

and

(G(1)

�↵

(·, 0))1↵N

, respectively. Finally, when writing convolution integrals, we will

sometimes omit the Latin and Greek indices.

Taking a radius r min{r1

/4, R0

/2, 1/8}, with R0

given by Theorem A.13 and

r1

by Proposition A.21, we write G�

as a perturbation of G(1) by letting

(A.78) w = G�

� T2r

G�

�G(1)

�

,

where the equality is understood to hold in W 1,p(B2r

;RN), and we take x0

= 0 in

the definition of the perturbation operator T2r

. Since r r0

/2, by Proposition A.21,

kT2r

kp

< 1/2. Thus, as in the proof of Theorem A.13, we can write

(A.79) G�

= G(1)

�

+1X

l=1

T l

2r

G(1)

�

+1X

l=0

T l

2r

w.

Claim. kwk⇤1,q;B2r

Cr1�n/p.

Proof of the claim. The proof is very similar to that of Lemma A.15. Therefore we

will only emphasize the necessary modifications and sketch the rest of the argument.

As in the proof of Lemma A.15, we estimate separately the norm of w over Br

and

B2r

\Br

. For kwk⇤1,q;B

r

, notice that straightforward computation using the definition

of T2r

shows that L(1)w = 0 on B2r

. Thus we have

(A.80) kwk⇤1,q;B

r

Ckwk⇤1,p;B2r

C�kG(1)

�

k⇤1,p;B2r

+ kG�

k⇤1,p;B2r

+ kT2r

G�

k⇤1,p;B2r

�

.

From the proof of Lemma A.15 (see (A.26)) we have kG(1)

�

k⇤1,p;B2r

Cr1�n/p. A similar

argument using Proposition A.4 shows that the same bound holds for kG�

k⇤1,p;B2r

and thus for kT2r

G�

k⇤1,p;B2r

as well, since T2r

is a bounded operator with respect to

k · k⇤1,p;B2r

. Therefore we obtain kwk⇤1,q;B

r

Cr1�n/p. and it remains to show that

kwk⇤1,q;B2r\Br

Cr1�n/p as well.

To estimate kwk1,q;B2r\Br

, note that, by definition and the triangle inequality,

(A.81) kwk⇤1,q;B2r\Br

kG�

k⇤1,q;B2r\Br

+ kG(1)

�

k⇤1,q;B2r\Br

+ kT2r

G�

k⇤1,q;B2r\Br

.

The second term on the right-hand side was already handled in the proof of Lemma

A.15, and the conclusion was that kG(1)

�

k⇤1,q;B2r\Br

Cr1�n/p. The same argument

yields the bound kG�

k⇤1,q;B2r\Br

Cr1�n/p.

The last term on the right-hand side of (A.81) requires some care, as the estimates

we have for G(1) are not as precise as the ones we have for E. For convenience, we

denote the two integrals in the definition of T2r

G�

by T I

2r

G�

and T II

2r

G�

, respectively.

A. THE FUNDAMENTAL SOLUTION OF DIVERGENCE-FORM ELLIPTIC SYSTEMS 115

For T I

2r

G�

, by definition we have

T I

2r

G�

(x) =�ˆB2r\B

r/2

@(2)i

G(1)

�

(x, y)B(y)G�

(y)dy

�ˆB

r/2

@(2)i

G(1)

�

(x, y)B(y)G�

(y)dy

⌘ '1

(x) + '2

(x).

By Proposition A.17 applied with F i

↵

(y) = Bi

↵�

(y)G��

(y)�B2r\B

r/2(y), and with the

help of Theorem A.2, we have k'1

k1,q;B1 C⇤kG

�

k1,q;B2r\B

r/2. Thus we infer that

k'1

k⇤1,q;B2r\Br

Cr1�n/qk'1

k1,q;B1 Cr1�n/qkG

�

k1,q;B2r\B

r/2,

and the right-most term has been shown above to be bounded by Cr1�n/p.

As for '2

, note that again by Proposition A.17, this time applied with F i

↵

(y) =

Bi

↵�

(y)G��

(y)�B

r/2(y), the function '

2

is a weak solution to L(1)u = 0 on B4r

\Br/2

.

Thus, by Theorems A.1 and A.2 and Proposition A.6(1),

k'2

k⇤1,q;B2r\Br

Ck'2

k⇤1,p;B3r\B2r/3

Cr1�n/pk'2

k1,p;B1

C⇤r1�n/pkG�

k1,p;B

r/2 Cr1�n/p.

Combining the estimates for '1

and '2

, we arrive at

(A.82) kT I

2r

G�

k⇤1,q;B2r\Br

Cr1�n/p.

By similar reasoning, we get kT II

2r

G�

k⇤1,q;B2r\Br

Cr1�n/p and hence kT2r

G�

k⇤1,q;B2r\Br

Cr1�n/p.

In view of (A.81) and the estimates derived above, we have kwk⇤1,q;B2r\Br

Cr1�n/p, and the claim is proved upon recalling kwk⇤

1,q;B

r

Cr1�n/p. ⇤

From the claim and our choice of r, we see that

(A.83)

�

�

�

�

�

1X

l=0

T l

2r

w

�

�

�

�

�

1,q;B2r

1X

l=0

kT2r

klq

kwk1,q;B2r Cr1�n/p.

Hence, as before, the Sobolev embedding then yields

(A.84)

�

�

�

�

�

1X

l=0

T l

2r

w

�

�

�

�

�

0;B2r

Cr1�n/p = Cr1+µ�n.

We next treat the term1P

l=1

T l

2r

G(1)

�

. Following the notation in the proof of Theorem

A.13, we write ul

for T l

2r

G(1)

�

. By definition, we have

ul,�

(x) = �ˆB2r

@(2)i

G(1)

↵�

(x, y)Bi

↵�

(y)ul�1,�

(y)dy(A.85)

A. THE FUNDAMENTAL SOLUTION OF DIVERGENCE-FORM ELLIPTIC SYSTEMS 116

�K

ˆB2r

G(1)

↵�

(x, y)ul�1,↵

(y)dy.

Claim. For l = 0, 1, 2, · · · , and x 2 B2r

\ {0} the following estimate holds.

(A.86) |ul

(x)| C(Crµ)l|x|2�n.

Proof of the claim. For l = 0, ul

reduces to G(1)

�

and (A.86) follows from Theorem

A.13 and the fact that r < R0

/2. Next, assuming (A.86) for l � 1, by the recursive

relation (A.86) we obtain

|(ul

)�

(x)| ⇤

ˆB2r

|rG(1)

�

(x, y)||ul�1

(y)|dy

+K

ˆB2r

|G(1)

�

(x, y)||ul�1

(y)|dy

C⇤

ˆB2r

(|x� y|1�n + |x� y|�1R1+µ�n

0

)(Crµ)l�1|y|2�ndy

+K

ˆB2r

(|x� y|2�n +R1+µ�n

0

)(Crµ)l�1|y|2�ndy

C(Crµ)l�1

ˆB2r

(⇤+Kr)|x� y|1�n|y|2�ndy,

where in the last step we used the inequalities r R0

/2 < R0

and |x � y| 4r.

Recalling (A.39), we infer from the above string of inequalities that

(A.87) |(ul

)�

(x)| C(Crµ)l�1|x|3�n C(Crµ)l|x|2�n,

where in the second inequality above we estimated |x|3�n by |x|2�nrµ. This completes

the induction step, and the claim is proved. ⇤

We now conclude the proof of Theorem A.8. Choosing

R1

= min{R0

/2, 1/8, r1

/4, (2C)1/µ},we see from (A.86) that |u

l

(x)| C2�l|x|2�n. Summing from l = 1 to 1, we get

(A.88)

�

�

�

�

�

1X

l=1

ul

(x)

�

�

�

�

�

C|x|2�n.

Recalling (A.79) and (A.84), we conclude that,

(A.89) |G�

(x)| C|x|2�n + Cr1+µ�n for r R1

and x 2 B2r

\ {0},which is precisely the desired conclusion. ⇤

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