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003 Floer homology and the heat flow

Dietmar A. Salamon Joa Weber

ETH-Zurich

23 April 2003

Abstract

The main theorem of this paper asserts that the Floer homology ofthe cotangent bundle of a closed Riemannian manifold M is naturallyisomorphic to the homology of the loop space. The idea of the proof is torelate the Floer equations in the cotangent bundle to the heat flow on theloop space.

1 Introduction

Let M be a compact Riemannian manifold and denote by LM the free loopspace. Consider the classical action functional

SV (x) =

∫ 1

0

(1

2|x(t)|2 − V (t, x(t))

)dt

for x : S1 → M . Here and throughout we identify S1 = R/Z and think ofx ∈ C∞(S1,M) as a smooth function x : R → M which satisfies x(t+1) = x(t).The critical points of SV are the 1-periodic solutions of the ode

∇tx = −∇Vt(x), (1)

where ∇V denotes the gradient with respect to the x-variable, and ∇tx denotesthe Levi-Civita connection. Let P = P(V ) denote the set of 1-periodic solutionsx : S1 → M of (1). In the case V = 0 these are the closed geodesics. Via theLegendre transformation the solutions of (1) can also be interpreted as thecritical points of the symplectic action AV : LT ∗M → R given by

AV (z) =

∫ 1

0

(〈y(t), x(t)〉 −H(t, x(t), y(t))

)dt

where z = (x, y) : S1 → T ∗M and the Hamiltonian H = HV : S1 × T ∗M → R

is given by

H(t, x, y) =1

2|y|2 + V (t, x) (2)

1

for y ∈ T ∗xM . A loop z(t) = (x(t), y(t)) in T ∗M is a critical point of AV iff

x is a solution of (1) and y(t) ∈ T ∗x(t)M is related to x(t) ∈ Tx(t)M via the

isomorphism TM → T ∗M induced by the Riemannian metric. For such loopsz the symplectic action AV (z) agrees with the classical action SV (x).

The negative L2 gradient flow of the classical action gives rise to a Morse-Witten complex which computes the homology of the loop space. For a realnumber a > 0 we shall denote by HMa

∗ (LM,SV ) the homology of the Morse-Witten complex of the functional SV corresponding to the solutions of (1) withSV (x) ≤ a. Here we assume that SV is a Morse function and its gradient flowsatisfies the Morse-Smale condition (i.e. the stable and unstable manifolds inter-sect transversally, see [2] for the unstable manifold). As in the finite dimensionalcase one can show that the Morse-Witten homology HMa

∗ (LM,SV ) is naturallyisomorphic to the singular homology of the sublevel set

LaM = x ∈ LM | SV (x) ≤ a .

On the other hand one can use the L2 gradient flow of the symplectic actionAV to construct Floer homology groups HF a

∗ (T∗M,HV ). Our main result is

the following

Theorem 1.1. There is a natural isomorphism

HF a∗ (T

∗M,HV ;Z) ∼= HMa∗ (LM,SV ;Z),

where HV : S1 × T ∗M → R is given by (2). If M is not simply connected thenthere is a separate isomorphism for each component of the loop space.

Corollary 1.2. There is a natural isomorphism

HF a∗ (T

∗M,HV ;Z) ∼= H∗(LaM ;Z),

where HV : S1 × T ∗M → R is given by (2). If M is not simply connected thenthere is a separate isomorphism for each component of the loop space.

Proof. Theorem 1.1 and Theorem A.7

Both the Morse-Witten homology HMa∗ (LM,SV ) and the Floer homology

HF a∗ (T

∗M,HV ) are based on the same chain complex Ca∗ which is generated

by the solutions of (1) and graded by the Morse index (as critical points ofSV ). In [21] it is shown that this Morse index agrees, up to a universal additiveconstant zero or one, with minus the Conley-Zehnder index. Thus it remainsto compare the boundary operators and this will be done by considering anadiabatic limit with a family of metrics on T ∗M which scales the vertical partdown to zero. Another approach to Corollary 1.2 is contained in Viterbo’spreprint [19].

2

The Floer chain complex and its adiabatic limit

We assume throughout that SV is a Morse function on the loop space, i.e. thatthe 1-periodic solutions of (1) are all nondegenerate. (For a proof that this holdsfor a generic potential V see [21].) Under this assumption

Pa(V ) := x ∈ P(V ) | SV (x) ≤ a

is a finite set for every real number a. Moreover, each critical point x ∈ P(V )has well defined stable and unstable manifolds with respect to the (negative)L2 gradient flow (see for example Davies [2]). Call SV Morse–Smale if it is aMorse function and the unstable manifold Wu(y) intersects the stable manifoldW s(x) transversally for any two critical points x, y ∈ P(V ). That this conditionholds for a generic perturbation V is proved in [22] (see Theorem A.6).

Assume SV is a Morse function and consider the Z-module

Ca = Ca(V ) =⊕

x∈Pa(V )

Zx.

If SV and AV are Morse–Smale then this module carries two boundary opera-tors. The first is defined by counting the (negative) gradient flow lines of SV .They are solutions u : R× S1 → M of the heat equation

∂su−∇t∂tu−∇Vt(u) = 0 (3)

satisfyinglim

s→±∞u(s, t) = x±(t), lim

s→±∞∂su = 0, (4)

where x± ∈ P(V ). The limits are uniform in t. The space of solutions of (3)and (4) will be denoted by M0(x−, x+;V ). The Morse–Smale hypothesis guar-antees that, for every pair x± ∈ Pa(V ), the space M0(x−, x+;V ) is a smoothmanifold whose dimension is equal to the difference of the Morse indices. In thecase of Morse index difference one it follows that the quotient M0(x−, x+;V )/Rby the (free) time shift action is a finite set. Counting the number of solutionswith appropriate signs gives rise to a boundary operator on Ca(V ). The ho-mology HMa

∗(LM,SV ) of the resulting chain complex is naturally isomorphic tothe singular homology of the loop space for every regular value a of SV :

HMa∗(LM,SV ) ∼= H∗(LaM ;Z), LaM := x ∈ LM | SV (x) ≤ a .

The details of this isomorphism will be established in a separate paper (seeAppendix A for a summary of the relevant results).

The second boundary operator is defined by counting the negative gradientflow lines of the symplectic action functional AV . These are the solutions (u, v) :R× S1 → TM of the Floer equations

∂su−∇tv −∇Vt(u) = 0, ∇sv + ∂tu− v = 0, (5)

lims→±∞

u(s, t) = x±(t), lims→±∞

v(s, t) = x±(t). (6)

3

For notational simplicity we identify the tangent and cotangent bundles of Mvia the metric. Counting the index-1 solutions of (5) and (6) with appropri-ate signs we obtain the Floer boundary operator. We wish to prove that theresulting Floer homology groups HFa

∗(T∗M,HV ) are naturally isomorphic to

HMa∗(LM,SV ). To construct this isomorphism we modify equation (5) by in-

troducing a small parameter ε as follows

∂su−∇tv −∇V (t, u) = 0, ∇sv + ε−2(∂tu− v) = 0. (7)

The space of solutions of (7) and (6) will be denoted by Mε(x−, x+;V ). TheFloer homology groups for different values of ε are isomorphic (see Remark 1.3below). Thus the task at hand is to prove that, for ε > 0 sufficiently small,there is a one-to-one correspondence between the solutions of (3) and thoseof (7). A first indication, why one might expect such a correspondence, is theenergy identity

Eε(u, v) =1

2

∫ ∞

−∞

∫ 1

0

(|∂su|2 + |∇tv +∇Vt(u)|2 + ε2|∇sv|2 + ε−2|∂tu− v|2

)

= SV (x−)− SV (x

+)

for the solutions of (7) and (6). It shows that ∂tu − v must converge to zeroin the L2 norm as ε tends to zero. If ∂tu = v then the first equation in (7) isequivalent to (3).

Remark 1.3. Let M be a Riemannian manifold. Then the tangent space ofthe cotangent bundle T ∗M at a point (x, y) with y ∈ T ∗

xM can be identifiedwith the direct sum TxM ⊕ T ∗

xM . The isomorphism takes the derivative z(t)of a curve R → T ∗M : t 7→ z(t) = (x(t), y(t)) to the pair (x(t),∇ty(t)). Withthis identification the almost complex structure Jε and the metric Gε on T ∗M ,given by

Jε =

(0 −εg−1

ε−1g 0

), Gε =

(ε−1g 00 εg−1

),

are compatible with the standard symplectic form ω on T ∗M . Here we denoteby g : TM → T ∗M the isomorphism induced by the metric. The case ε = 1corresponds to the standard almost complex structure. The Floer equations forthe almost complex structure Jε and the Hamiltonian (2) are

∂sw − Jε(w)(∂tw −XHt(w)) = 0.

If we write w(s, t) = (u(s, t), v(s, t)) with v(s, t) ∈ T ∗u(s,t)M then this equation

has the form

∂su− εg−1∇tv − ε∇Vt(u) = 0, ∇sv + ε−1g∂tu− ε−1v = 0. (8)

Now define

u(s, t) := u(ε−1s, t), v(s, t) := g−1v(ε−1s, t).

4

Then (u, v) is a solution of (8) if and only if (u, v) is a solution of (7). In viewof this discussion it follows from the Floer homotopy argument that the Floerhomology defined with the solutions of (7) is independent of the choice of ε > 0.

Assume SV is Morse–Smale. Then we shall prove that, for every a ∈ R, thereexists an ε0 > 0 such that, for 0 < ε < ε0 and every pair x+, x− ∈ Pa(V ) withMorse index difference one, there is a natural bijective correspondence betweenthe (shift equivalence classes of) solutions of (3), (4) and those of (7), (6). Thiswill follow from Theorems 3.1 and 9.1 below. The remainder of this paper isdevoted to the proof of this result.

The general outline of the proof is similar to that of the Atiyah–Floer conjec-ture in [4] which compares two elliptic pdes via an adiabatic limit argument. Bycontrast our adiabatic limit theorem compares elliptic with parabolic equations.Thie leads to new features in the analysis that are related to the fact that theparabolic equation requires different scaling in space and time directions.

The present paper is organized as follows. The next section explains therelevant linearized operators and states the estimates for the right inverse.These are proved in Appendices B and C. In Section 3 we construct a mapT ε : M0(x−, x+;V ) → Mε(x−, x+;V ) which assigns to every parabolic cylin-der of index one a nearby Floer connecting orbit for ε > 0 sufficiently small.The existence of this map was established in the thesis of the second author [20],where the results of Section 2, Section 3, and Appendix C were proved. Sec-tions 4, 5, and 6 are of preparatory nature and establish uniform estimates forthe solutions of (7). Section 7 deals with exponential decay, Section 8 estab-lishes local surjectivity of the map T ε by a time-shift argument, and in Section 9we prove that T ε is bijective. Things are put together in Section 10 where wecompare orientations and prove Theorem 1.1. Appendix A summarizes someresults about the heat flow (3) which will be proved in [22].

2 The linearized operators

Linearizing the heat equation (3) gives rise to the operator

D0u : Ω0(R× S1, u∗TM) → Ω0(R× S1, u∗TM)

given byD0

uξ = ∇sξ −∇t∇tξ −R(ξ, ∂tu)∂tu−∇ξ∇V (t, u), (9)

for every element ξ of the set Ω0(R× S1, u∗TM) of smooth vector fields alongu. Since SV is Morse, this is a Fredholm operator between appropriate Sobolevcompletions. More precisely, define Lu = Lp

u and Wu = Wpu as the completions

of the space of smooth compactly supported sections of the pullback tangentbundle u∗TM → R× S1 with respect to the norms

‖ξ‖0,p =

(∫ ∞

−∞

∫ 1

0

|ξ|p dtds)1/p

,

5

‖ξ‖1,p =

(∫ ∞

−∞

∫ 1

0

|ξ|p + |∇sξ|p + |∇t∇tξ|p dtds)1/p

.

Then D0u : Wp

u → Lpu is a Fredholm operator for p > 1 (Theorem A.3) with

indexindexD0

u = indV (x−)− indV (x

+),

where indV (x) denotes the Morse index, i.e. the number of negative eigenvaluesof the Hessian

H0xξ = −∇t∇tξ −R(ξ, x)x−∇ξ∇Vt(x).

(Here R denotes the Riemann curvature tensor.) The Morse–Smale conditionasserts that the operator D0

u is surjective for every finite energy solution of (3).That this condition can be achieved by a generic perturbation of V is provedin [22] (see Appendix A).

Linearizing equation (7) gives rise to the first order differential operator

Dεu,v : W 1,p(R× S1, u∗TM ⊕ u∗TM) → Lp(R× S1, u∗TM ⊕ u∗TM)

given by

Dεu,v

(ξη

)=

(∇sξ −∇tη −R(ξ, ∂tu)v −∇ξ∇Vt(u)∇sη +R(ξ, ∂su)v + ε−2(∇tξ − η)

). (10)

Remark 2.1. Assume SV is Morse and let p > 1. Then Dεu,v is a Fredholm

operator for every pair (u, v) that satisfies (6) and its index is given by

indexDεu,v = indV (x

−)− indV (x+).

To see this rescale u and v as in Remark 1.3. Then the operator on the rescaledvector fields ξ(s, t) := ξ(ε−1s, t) and η(s, t) := g−1η(ε−1s, t) has the same formas in Floer’s original papers [6] with the almost complex structure Jε of Re-mark 1.3. That this operator is Fredholm was proved in [5, 17, 15] for p = 2.An elegant proof of the Fredholm property for general p > 1 was given by Don-aldson [3] for the instanton case; it adapts easily to the symplectic case [16].The Fredholm index can be expressed as a difference of the Conley–Zehnderindices [17, 15]. That it agrees with the difference of the Morse indices wasproved in [20].

Let us now fix a solution u of (3) and define v := ∂tu. For this pair (u, v)we must prove that the operator Dε

u := Dεu,∂tu

is onto for ε > 0 sufficientlysmall and prove an estimate for the right inverse which is independent of ε.We will establish this under the assumption that the operator D0

u is onto. Toobtain uniform estimates for the inverse with constants independent of ε wemust work with suitable ε-dependent norms. For compactly supported vectorfields ζ = (ξ, η) ∈ Ω0(R× S1, u∗TM ⊕ u∗TM) define

‖ζ‖0,p,ε =(∫ ∞

−∞

∫ 1

0

(|ξ|p + εp |η|p) dtds)1/p

,

6

‖ζ‖1,p,ε =(∫ ∞

−∞

∫ 1

0

(|ξ|p + εp |η|p + εp |∇tξ|p + ε2p |∇tη|p

+ ε2p |∇sξ|p + ε3p |∇sη|p)dtds

)1/p

.

Theorem 2.2. Let (u, v) : R× S1 → TM be a smooth map such that

sups,t

|v(s, t)| < ∞,

the derivatives ∂su, ∂tu,∇t∂su,∇t∂tu are bounded and lims→±∞ u(s, t) exists,uniformly in t. Then, for every p > 1, there are positive constants c and ε0 suchthat, for every ε ∈ (0, ε0) and every ζ = (ξ, η) ∈ W 1,p(R×S1, u∗TM ⊕ u∗TM),we have

ε−1 ‖∇tξ − η‖Lp + ‖∇tη‖Lp + ‖∇sξ‖Lp + ε ‖∇sη‖Lp

≤ c(∥∥Dε

u,vζ∥∥0,p,ε

+ ‖ξ‖Lp + ε2 ‖η‖Lp

).

(11)

The formal adjoint operator (Dεu,v)

∗ defined below satisfies the same estimate.Moreover, the constants c and ε0 are invariant under s-shifts of u.

The formal adjoint operator

(Dεu,v)

∗ : W 2,p(R× S1, u∗TM ⊕ u∗TM) → W 1,p(R× S1, u∗TM ⊕ u∗TM)

with respect to the (0, 2, ε)-inner product associated to the (0, 2, ε)-norm hasthe form

(Dεu,v)

∗

(ξη

)=

(−∇sξ −∇tη −R(ξ, v)∂tu−∇ξ∇Vt(u) + ε2R(η, v)∂su

−∇sη + ε−2(∇tξ − η)

)

for ξ, η ∈ W 1,p(R× S1, u∗TM). We shall also use the projection operator

πε : Lp(S1, u∗TM)× Lp(S1, u∗TM) → W 1,p(S1, u∗TM)

given byπε(ξ, η) = (1l− ε∇t∇t)

−1(ξ − ε2∇tη).

This operator will be applied to the pair (ξ(s, ·), η(s, ·)) for every s.

Theorem 2.3. Assume SV is Morse-Smale and let u ∈ M0(x−, x+;V ).Then,for every p > 1, there are positive constants c and ε0 (invariant under s-shiftsof u) such that, for every ε ∈ (0, ε0) the following are true. The operatorDε

u := Dεu,∂tu

is onto and for every pair

ζ := (ξ, η) ∈ im (Dεu)

∗ ⊂ W 1,p(R× S1, u∗TM ⊕ u∗TM)

we have

‖ξ‖Lp + ε1/2 ‖η‖Lp + ε1/2 ‖∇tξ‖Lp ≤ c(ε ‖Dε

uζ‖0,p,ε + ‖πε(Dεuζ)‖Lp

)(12)

and‖ζ‖1,p,ε ≤ c

(ε ‖Dε

uζ‖0,p,ε + ‖πε(Dεuζ)‖Lp

). (13)

7

The proofs of Theorems 2.2 and 2.3 are given in Appendix C. They arebased on a simplified form of Theorem 2.2 for flat manifolds with V = 0 whichis proved in Appendix B. In particular, Corollary B.3 shows that the ε-weightson the left hand side of equation (11) appear in a natural manner by a rescalingargument and, for p = 2, these terms can be interpreted as a linearized versionof the energy. This was in fact the motivation for introducing the above ε-dependent norms. The proof of Theorem 2.3 is based on Theorem 2.2 and acomparison of the operators D0

u and Dεu.

To construct a solution of (7) near a parabolic cylinder it is useful to combineTheorems 2.2 and 2.3 into the following corollary. This corollary involves anε-dependent norm which at first glance appears to be somewhat less natural butplays a useful role for technical reasons.

Given a smooth map u : R × S1 → M and a compactly supported pair ofvector fields ζ = (ξ, η) ∈ Ω0(R× S1, u∗TM ⊕ u∗TM) we define

|||ζ|||ε := ‖ξ‖p + ε1/2 ‖η‖p + ε1/2 ‖∇tξ‖p + ‖η −∇tξ‖p + ε2 ‖∇sη‖p+ ε ‖∇tη‖p + ε ‖∇sξ‖p + ε3/2p ‖ξ‖∞ + ε1/2+2/p ‖η‖∞ .

(14)

For small ε this norm is much bigger than the (1, p, ε)-norm.

Corollary 2.4. Assume SV is Morse-Smale and let u ∈ M0(x−, x+;V ). Then,for every p > 1, there are positive constants c and ε0 such that, for everyε ∈ (0, ε0) the following holds. If

ζ = (ξ, η) ∈ im (Dεu)

∗, ζ′ = (ξ′, η′) := Dεuζ,

then|||ζ|||ε ≤ c

(‖ξ′‖p + ε3/2 ‖η′‖p

). (15)

Proof. Let c2 be the constant of Theorem 2.2 and c3 be the constant of Theo-rem 2.3. Then, by Theorem 2.3,

‖ξ‖p + ε1/2 ‖η‖p + ε1/2 ‖∇tξ‖p≤ c3

(ε ‖ξ′‖p + ε2 ‖η′‖p +

∥∥(1l− ε∇t∇t)−1(ξ′ − ε2∇tη

′)∥∥p

)

≤ c3

((1 + ε) ‖ξ′‖p + (ε2 + κpε

3/2) ‖η′‖p)

≤ c4

(‖ξ′‖p + ε3/2 ‖η′‖p

).

Here the second step follows from Lemma C.3. Combining the last estimatewith Theorem 2.2 we obtain

‖η −∇tξ‖p + ε ‖∇tη‖p + ε ‖∇sξ‖p + ε2 ‖∇sη‖p≤ c2ε

(‖ξ′‖p + ε ‖η′‖p + ‖ξ‖p + ε2 ‖η‖p

)

≤ c2

(ε ‖ξ′‖p + ε2 ‖η′‖p + c4ε

(‖ξ′‖p + ε3/2 ‖η′‖p

))

≤ c2(1 + c4)(ε ‖ξ′‖p + ε2 ‖η′‖p

).

8

Now let c5 be the constant of Lemma 2.5 below. Then

ε3/2p ‖ξ‖∞ ≤ c5

(‖ξ‖p + ε1/2 ‖∇tξ‖p + ε ‖∇sξ‖p

),

ε1/2+2/p ‖η‖∞ ≤ c5

(ε1/2 ‖η‖p + ε ‖∇tη‖p + ε2 ‖∇sη‖p

).

(16)

(Here we used the cases (β1, β2) = (1, 2) and (β1, β2) = (1/2, 1).) Combiningthese four estimates we obtain (15).

The second estimate in the proof of Corollary 2.4 shows that one can obtain astronger estimate than (15) from Theorems 2.2 and 2.3. Namely, (15) continuesto hold if |||ζ|||ε is replaced by the stronger norm where the Lp norms of ∇tξ− η,∇tη, ∇sξ, and ∇sη are multiplied by an additional factor ε−1/2. The reason fornot using this stronger norm lies in the proof of Theorem 3.1. In the first stepof the iteration we solve an equation of the form Dε

uζ0 = ζ′ = (0, η′) where η′ isbounded (in Lp) with all its derivatives. Our goal in this first step is to obtainthe sharpest possible estimate for ζ0 and its first derivatives. We shall see thatthis estimate has the form |||ζ0|||ε ≤ cε2 and that such an estimate in terms ofε2 cannot be obtained with the stronger norm indicated above.

Lemma 2.5. Let u ∈ C∞(R×S1,M) such that ‖∂su‖∞ and ‖∂tu‖∞ are finiteand lims→±∞ u(s, t) exists, uniformly in t. Then, for every p > 2, there is aconstant c > 0 such that

‖ξ‖∞ ≤ cε−(β1+β2)/p(‖ξ‖p + εβ1 ‖∇tξ‖p + εβ2 ‖∇sξ‖p

)

for every ε ∈ (0, 1], every pair of nonnegative real numbers β1 and β2, and everycompactly supported vector field ξ ∈ Ω0(R× S1, u∗TM).

Proof. Define u : Zε := R×(R/ε−β1Z

)→ M and ξ ∈ Ω0(Zε, u

∗TM) by

u(s, t) := u(εβ2s, εβ1t), ξ(s, t) := ξ(εβ2s, εβ1t).

The estimate is equivalent to the Sobolev inequality

∥∥ξ∥∥∞

≤ c(∥∥ξ

∥∥p+∥∥∇tξ

∥∥p+∥∥∇sξ

∥∥p

)

with a uniform constant c = c(p, ‖∂su‖∞ , ‖∂tu‖∞) that is independent of ε ∈(0, 1]. (To see how the L∞ bounds on ∂su and ∂tu enter the estimate, embeddM into some euclidean space and use the Gauss-Weingarten formula.)

3 Existence and uniqueness

In the next theorem we denote by

Φ(x, ξ) : TxM → Texpx(ξ)M

parallel transport along the geodesic τ 7→ expx(τξ).

9

Theorem 3.1 (Existence). Assume SV is Morse–Smale and fix two constantsa ∈ R and p > 2. Then there are positive constants c and ε0 such that, for everyε ∈ (0, ε0) and every pair x± ∈ Pa(V ) of index difference one, there is a shiftequivariant map

T ε : M0(x−, x+;V ) → Mε(x−, x+;V )

with the following properties. If u ∈ M0(x−, x+;V ) then

T ε(u) = (uε, vε), uε = expu(ξ), vε = Φ(u, ξ)(∂tu+ η),

where the pair (ξ, η) ∈ im (Dεu)

∗ satisfies the inequalities

‖∇tξ − η‖Lp + ‖ξ‖Lp + ε1/2 ‖η‖Lp + ε1/2 ‖∇tξ‖Lp

+ ε ‖∇tη‖Lp + ε ‖∇sξ‖Lp + ε2 ‖∇sη‖Lp ≤ cε2(17)

and‖ξ‖L∞ ≤ cε2−3/2p, ‖η‖L∞ ≤ cε3/2−2/p. (18)

Remark 3.2. The estimates (17) and (18) can be summarized in the form

|||ζ|||ε ≤ cε2.

(with a larger constant c).

Theorem 3.3 (Uniqueness). Assume SV is Morse–Smale and fix two con-stants a ∈ R and C > 0. Then there are positive constants δ and ε0 such that,for every ε ∈ (0, ε0), every pair x± ∈ Pa(V ) of index difference one, and everyu ∈ M0(x−, x+;V ) the following holds. If

(ξ, η) ∈ im (Dεu)

∗, ‖ξ‖L∞ ≤ δε1/2, ‖η‖L∞ ≤ C, (19)

and the pairuε := expu(ξ), vε := Φ(u, ξ)(∂tu+ η),

belongs to the moduli space Mε(x−, x+;V ), then (uε, vε) = T ε(u).

In the hypotheses of Theorem 3.3 we did not specify the Sobolev space towhich ζ = (ξ, η) is required to belong. The reason is that it is smooth and, byexponential decay, belongs to the Sobolev space W k,p(R× S1, u∗TM ⊕ u∗TM)for every integer k ≥ 0 and every p ≥ 1.

The proof of Theorem 3.1 is based on the Newton–Picard iteration methodto detect a zero of a map near an approximate zero. The first step is to define asuitable map between Banach spaces. In order to do so let (u, v) : R×S1 → TMbe a smooth map and consider the map Fε

u,v : W 1,p(R×S1, u∗TM ⊕u∗TM) →Lp(R× S1, u∗TM ⊕ u∗TM) given by

Fεu,v

(ξη

):=

(Φ(u, ξ)−1 0

0 Φ(u, ξ)−1

)Fε

(expu ξ

Φ(u, ξ)(v + η)

), (20)

10

where

Fε

(uε

vε

):=

(∂su

ε −∇tvε −∇Vt(u

ε)∇sv

ε + ε−2(∂tuε − vε)

). (21)

Thus, abbreviating Φ := Φ(u, ξ), we have

Fεu,v

(ξη

):=

(Φ−1 (∂s expu(ξ) −∇t(Φ(v + η)) −∇Vt(expu(ξ)))Φ−1

(∇s(Φ(v + η)) + ε−2∂t expu(ξ)

)− ε−2(v + η)

).

Moreover, the differential of Fεu,v at the origin is given by

dFεu,v(0, 0) = Dε

u,v

(see [20, Appendix A.3]).One of the key ingredients in the iteration is to have control over the variation

of derivatives. This is provided by the following quadratic estimates.

Proposition 3.4. There exists a constant δ > 0 with the following significance.For every p > 1 and every c0 > 0 there is a constant c > 0 such that the followingis true. Let (u, v) : R×S1 → TM be a smooth map and Z = (X,Y ), ζ = (ξ, η) ∈Ω0(R× S1, u∗TM ⊕ u∗TM) be two pairs of vector fields along u such that

‖∂su‖∞ + ‖∂tu‖∞ + ‖v‖∞ ≤ c0, ‖ξ‖∞ + ‖X‖∞ ≤ δ, ‖η‖∞ + ‖Y ‖∞ ≤ c0.

Then the vector fields F1, F2 along u, defined by

Fεu,v(Z + ζ) −Fε

u,v(Z)− dFεu,v(Z)ζ =:

(F1

F2

),

satisfy the inequalities

‖F1‖p ≤ c‖ξ‖∞(‖ξ‖p + ‖η‖p + ‖∇tξ‖p + ‖∇sξ‖p‖ξ‖∞

)

+ c(‖∇tX‖p + ‖∇sX‖p

)‖ξ‖2∞ + c‖∇tX‖p‖ξ‖∞‖η‖∞

+ c‖X‖∞(‖∇sξ‖p‖ξ‖∞ + ‖∇tξ‖p‖η‖∞

),

‖F2‖p ≤ c‖ξ‖∞(ε−2‖ξ‖p + ‖η‖p + ‖∇sξ‖p + ε−2‖∇tξ‖p‖ξ‖∞

)

+ c(‖∇sX‖p + ε−2‖∇tX‖p

)‖ξ‖2∞ + c‖∇sX‖p‖ξ‖∞‖η‖∞

+ c‖X‖∞(ε−2‖∇tξ‖p‖ξ‖∞ + ‖∇sξ‖p‖η‖∞

).

Proposition 3.5. There exists a constant δ > 0 with the following significance.For every p > 1 and every c0 > 0 there is a constant c > 0 such that the followingis true. Let (u, v) : R×S1 → TM be a smooth map and Z = (X,Y ), ζ = (ξ, η) ∈Ω0(R× S1, u∗TM ⊕ u∗TM) be two pairs of vector fields along u such that

‖∂su‖∞ + ‖∂tu‖∞ + ‖v‖∞ ≤ c0, ‖X‖∞ ≤ δ, ‖Y ‖∞ ≤ c0.

11

Then the vector fields F1, F2 along u, defined by

dFεu,v(Z)ζ − dFε

u,v(0)ζ =:

(F1

F2

),

satisfy the inequalities

‖F1‖p ≤ c‖ξ‖∞(‖X‖p + ‖Y ‖p + ‖∇tX‖p + ‖∇sX‖p‖X‖∞

)

+ c‖X‖∞(‖η‖p + ‖∇tξ‖p + ‖∇sξ‖p‖X‖∞ + ‖∇tX‖p‖η‖∞

),

‖F2‖p ≤ c‖ξ‖∞(ε−2‖X‖p + ε−2‖∇tX‖p‖X‖∞ + ‖Y ‖p + ‖∇sX‖p

)

+ c‖X‖∞(ε−2‖∇tξ‖p‖X‖∞ + ‖η‖p + ‖∇sξ‖p + ‖∇sX‖p‖η‖∞

).

For the proof of Propositions 3.4 and 3.5 we refer to [20, Chapter 5]. Tounderstand the estimate of Proposition 3.5 note that η and Y appear only aszeroth order terms, that ∇sξ and ∇sX appear only in cubic terms in F1, andthat ∇tξ and ∇tX appear only in cubic terms in F2. This follows from the factthat the first component of Fε is linear in ∂su and the second component islinear in ∂tu. In Proposition 3.4 we have included cubic terms that arise whenthe derivative hits X . In this case we must use the L∞ norms on the factorsξ and η and can profit from the fact that ∇sX and ∇tX will be small in Lp.The constant δ appears as a condition for the pointwise quadratic estimates insuitable coordinate charts on M .

We now reformulate the quadratic estimates in terms of the norm (14).

Corollary 3.6. There exists a constant δ > 0 with the following significance.For every p > 1 and every c0 > 0 there is a constant c > 0 such that thefollowing holds. If (u, v), Z = (X,Y ) and ζ = (ξ, η) satisfy the hypotheses ofProposition 3.4 then

∥∥Fεu,v(Z + ζ)−Fε

u,v(Z)− dFεu,v(Z)ζ

∥∥0,p,ε3/2

≤ c|||ζ|||ε(ε−1/2 ‖ξ‖∞ + ε−1 ‖ξ‖2∞

)+ cε−1−3/2p|||Z|||ε|||ζ|||ε

(‖ξ‖∞ + ε1/2 ‖η‖∞

).

If (u, v), Z = (X,Y ) and ζ = (ξ, η) satisfy the hypotheses of Proposition 3.5then

∥∥dFεu,v(Z)ζ − dFε

u,v(0)ζ∥∥0,p,ε3/2

≤ c(ε−1/2−3/2p|||Z|||ε + ε−1−7/2p|||Z|||2ε

)|||ζ|||ε.

Proof. The result follows from Propositions 3.5 and 3.4 via term by term in-spection. In particular, we must use the inequalities

‖ξ‖∞ + ε ‖η‖∞ ≤ cε−3/p ‖ζ‖1,p,ε , ‖X‖∞ ≤ ε−3/2p|||Z|||ε

at various places. The first follows from Lemma 2.5 with (β1, β2) = (1, 2) andthe second from the definition of the norm in (14).

12

Proof of Theorem 3.1. Given u ∈ M0(x−, x+;V ) with x± ∈ Pa(V ) we aim todetect an element of Mε(x−, x+;V ) near u. We set v := ∂tu and carry out theNewton–Picard iteration method for the map Fε

u := Fεu,∂tu

. Key ingredientsare a small initial value, a uniformly bounded right inverse and control over thevariation of derivatives (which is provided by the quadratic estimates above).Because SV is Morse-Smale, the sets Pa(V ) and M0(x−, x+;V )/R are finite(the latter in addition relies on the assumption of index difference one). Allconstants appearing below turn out to be invariant under s-shifts of u. Hencethey can be chosen to depend on a only.

Since u ∈ M0(x−, x+;V ) it follows from Theorems A.1 and A.2 that thereis a constant c0 > 0 such that

‖∂su‖∞ + ‖∂tu‖∞ + ‖∇t∂tu‖∞ ≤ c0 (22)

and‖∇t∂su‖∞ + ‖∇t∂su‖p + ‖∇t∇t∂su‖p ≤ c0. (23)

Thus the assumptions in Theorem 2.2, Theorem 2.3, Proposition 3.4, Proposi-tion 3.5 and Lemma 2.5 are satisfied. Moreover, by (23) the value of the initialpoint Z0 := 0 is indeed small with respect to the (0, p, ε)-norm:

‖Fεu(0)‖0,p,ε = ‖Fε(u, ∂tu)‖0,p,ε =

∥∥∥∥(

0∇s∂tu

)∥∥∥∥0,p,ε

≤ c0ε. (24)

Here we used in addition (20), (21) and the parabolic equations. Define theinitial correction term ζ0 = (ξ0, η0) by

ζ0 := −Dεu∗(Dε

uDεu∗)−1Fε

u(0).

Recursively, for ν ∈ N, define the sequence of correction terms ζν = (ξν , ην) by

ζν := −Dεu∗(Dε

uDεu∗)−1Fε

u(Zν), Zν = (Xν , Yν) :=

ν−1∑

ℓ=0

ζℓ. (25)

We prove by induction that there is a constant c > 0 such that

|||ζν |||ε ≤c

2νε2, ‖Fε

u(Zν+1)‖0,p,ε3/2 ≤ c

2νε7/2−3/2p. (Hν)

Initial Step: ν = 0. By definition of ζ0 we have

Dεuζ0 = −Fε

u(0) =

(0

−∇s∂tu

).

Thus, by Theorem 2.3 (with constant c1 > 0),

‖ξ0‖p + ε1/2‖η0‖p + ε1/2‖∇tξ0‖p ≤ c1 (ε‖(0,∇s∂tu)‖0,p,ε + ‖πε(0,∇s∂tu)‖p)≤ c1

(ε2‖∇s∂tu‖p + ε2‖∇t∇s∂tu‖p

)

≤ c0c1ε2.

13

Here the second inequality follows from Lemma C.3 and the last from (23). ByTheorem 2.2 (with constant c2 > 0),

‖∇tξ0 − η0‖p + ε‖∇tη0‖p + ε‖∇sξ0‖p + ε2‖∇sη0‖p≤ c2ε

(‖(0,∇s∂tu)‖0,p,ε + ‖ξ0‖p + ε2‖η0‖p

)

≤ c2ε(ε‖∇s∂tu‖p + c0c1ε

2)

≤ c0c2(1 + c1ε)ε2.

The last inequality follows again from (23). Combining these two estimateswith (16) we obtain

ε3/2p ‖ξ0‖∞ + ε1/2+2/p ‖η0‖∞ ≤ |||ζ0|||ε ≤ cε2. (26)

with a suitable constant c > 0 (depending only on c0, c1, c2 and the constantof Lemma 2.5). This proves the first estimate in (Hν) for ν = 0. To prove thesecond estimate we observe that Z1 = ζ0 and hence, by Proposition 3.4 (withconstant c3 > 0),

‖Fεu(Z1)‖0,p,ε3/2

= ‖Fεu(ζ0)−Fε

u(0)−Dεuζ0‖0,p,ε3/2

≤ c3‖ξ0‖∞(‖ξ0‖p + ‖η0‖p + ‖∇tξ0‖p + ‖∇sξ0‖p‖ξ0‖∞

)

+ c3ε3/2‖ξ0‖∞

(ε−2‖ξ0‖p + ‖η0‖p + ‖∇sξ0‖p + ε−2‖∇tξ0‖p‖ξ0‖∞

)

≤ cε7/2−3/2p.

with a suitable constant c > 0 (depending only on c0, c1, c2 and the constant ofLemma 2.5). Thus we have proved (Hν) for ν = 0. From now on we fix theconstant c for which the estimate (H0) has been established.

Induction step: ν − 1 ⇒ ν. Let ν ≥ 1 and assume that (H0), . . . , (Hν−1)are true. Then

|||Zν |||ε ≤ν−1∑

ℓ=0

|||ζℓ|||ε ≤ cε2ν−1∑

ℓ=0

2−ℓ ≤ 2cε2,

‖Fεu(Zν)‖0,p,ε3/2 ≤ c

2ν−1ε7/2−3/2p.

By (25) we haveDε

uζν = −Fεu(Zν), ζν ∈ im(Dε

u)∗.

Hence, by Corollary 2.4, (with constant c4 > 0),

|||ζν |||ε ≤ c4 ‖Fεu(Zν)‖0,p,ε3/2 ≤ cc4

2ν−1ε7/2−3/2p ≤ c

2νε2. (27)

The last inequality holds whenever c4ε3/2−3/2p ≤ 1/2.

14

By what we have just proved the vector fields Zν and ζν satisfy the require-ments of Corollary 3.6 (with the constant c5 > 0). Hence

‖Fεu(Zν+1)‖0,p,ε3/2 ≤ ‖Fε

u(Zν + ζν)−Fεu(Zν)− dFε

u(Zν)ζν‖0,p,ε3/2+ ‖dFε

u(Zν)ζν −Dεuζν‖0,p,ε3/2

≤ c5

(ε−1/2 ‖ξν‖∞ + ε−1 ‖ξν‖2∞

)|||ζν |||ε

+ c5ε−1−3/2p|||Zν |||ε

(‖ξν‖∞ + ε1/2 ‖ην‖∞

)|||ζν |||ε

+ c5ε−1/2−3/2p|||Zν |||ε|||ζν |||ε + c5ε

−1−7/2p|||Zν |||2ε|||ζν |||ε≤ c5

(cε3/2−3/2p + c2ε3−3/p

)|||ζν |||ε + 2c2c5ε

3−7/2p|||ζν |||ε+ 2cc5ε

3/2−3/2p|||ζν |||ε + 4c2c5ε3−7/2p|||ζν |||ε

≤ 1

2c4|||ζν |||ε

≤ c

2νε7/2−3/2p.

In the third step we have used the inequalities ‖ξν‖∞ ≤ ε−3/2p|||ζν |||ε ≤ cε2−3/2p

and ‖ξν‖∞ + ε1/2 ‖ην‖∞ ≤ ε−2/p|||ζν |||ε ≤ cε2−2/p as well as |||Z|||ν ≤ 2cε2. Thefourth step holds for ε sufficiently small, and the last step follows from (27).This completes the induction and proves (Hν) for every ν.

It follows from (Hν) that Zν is a Cauchy sequence with respect to ||| · |||ε.Denote its limit by

ζ := limν→∞

Zν =∞∑

ν=0

ζν .

By construction and by (Hν), the limit satisfies

|||ζ|||ε ≤ 2cε2, Fεν (ζ) = 0, ζ ∈ im (Dε

u)∗.

Hence, by (20), the pair

(uε, vε) := (expu(ξ),Φ(u, ξ)(∂tu+ η))

is a solution of (7). Since |||ζ|||ε is finite it follows that (∂suε,∇sv

ε) is bounded.Hence, by the standard elliptic bootstrapping arguments for pseudoholomorphiccurves, the shifted functions uε(s+ ·, ·), vε(s+ ·, ·) converge in the C∞ topologyon every compact set as s tends to ±∞. Since ζ ∈ W 1,p, the limits must bethe periodic orbits x± and, moreover, the pair (∂su

ε(s, t),∇svε(s, t)) converges

to zero, uniformly in t, as s tends to ±∞. Hence (uε, vε) ∈ Mε(x−, x+;V ).Evidently, each step in the iteration including the constants in the estimates isinvariant under time shift. This proves the theorem.

Proof of Theorem 3.3. Fix a constant p > 2 and an index one parabolic cylinderu ∈ M0(x−, x+;V ). Denote v := ∂tu and Fε

u := Fεu,∂tu

. As in the proof ofTheorem 3.1, the map u satisfies the estimates (22) and (23). Denote by

T ε(u) = (expu(X),Φ(u,X)(∂tu+ Y ))

15

the solution of (7) constructed in Theorem 3.1. Then

Z ∈ im (Dεu)

∗, Fεu(Z) = 0, |||Z|||ε ≤ cε2

for a suitable constant c > 0. Now suppose (uε, vε) ∈ Mε(x−, x∗;V ) satisfiesthe hypotheses of the theorem. This means that there is a pair

ζ = (ξ, η) ∈ W 1,p(R× S1, u∗TM ⊕ u∗TM)

such that

ζ ∈ im (Dεu)

∗, Fεu(ζ) = 0, ‖ξ‖∞ ≤ δε1/2, ‖η‖∞ ≤ C.

The differenceζ′ := (ξ′, η′) := ζ − Z

satisfies the inequalities

‖ξ′‖∞ ≤ δε1/2 + cε2−3/2p ≤ 2δε1/2, ‖η′‖∞ ≤ C + cε3/2−2/p ≤ 2C,

provided that ε is sufficiently small. Hence, by Corollary 2.4 (with a constantc1 > 0) and Corollary 3.6 (with a constant c2 > 0), we have

|||ζ′|||ε ≤ c1 ‖Dεuζ

′‖0,p,ε3/2≤ c1 ‖Fε

u(Z + ζ′)−Fεu(Z)− dFε

u(Z)ζ′‖0,p,ε3/2+ c1 ‖dFε

u(Z)ζ′ − dFεu(0)ζ

′‖0,p,ε3/2

≤ c1c2

(ε−1/2 ‖ξ′‖∞ + ε−1 ‖ξ′‖2∞

)|||ζ′|||ε

+ c1c2ε−1−3/2p|||Z|||ε

(‖ξ′‖∞ + ε1/2 ‖η′‖∞

)|||ζ′|||ε

+ c1c2ε−1/2−3/2p|||Z|||ε|||ζ′|||ε + c1c2ε

−1−7/2p|||Z|||2ε|||ζ′|||ε≤ c1c2

(2δ + 4δ2

)|||ζ′|||ε + cc1c2ε

3/2−3/2p(2δ + 2C

)|||ζ′|||ε

+ cc1c2ε3/2−3/2p|||ζ′|||ε + c2c1c2ε

3−7/2p|||ζ′|||ε≤ 1

2|||ζ′|||ε.

The last inequality holds when δ and ε are sufficiently small. It follows thatζ′ = 0 and this proves the theorem.

4 An apriori estimate

Theorem 4.1. Fix a potential V : S1 ×M → R and a constant c0 > 0. Thenthere is a constant C = C(c0, V ) > 0 such that the following holds. If 0 < ε ≤ 1and (u, v) : R× S1 → TM is a solution of (7) such that

Eε(u, v) ≤ c0, sups∈R

AV (us, vs) ≤ c0 (28)

then ‖v‖∞ ≤ C.

16

For ε = 1 this result was proved by Cieliebak [1, Theorem 5.4]. His proofcombines the 2-dimensional maximum principle and the Krein-Rutman theorem.Our proof is based on the following L2 estimate.

Proposition 4.2. Fix a potential V : S1×M → R and a constant c0 > 0. Thenthere is a constant c = c(c0, V ) > 0 such that the following holds. If 0 < ε ≤ 1and (u, v) : R× S1 → TM is a solution of (7) that satisfies (28) then

sups∈R

∫ 1

0

|v(s, t)|2 dt ≤ c.

Lemma 4.3. Let f : [−T, T ] → R be a C2 function and δ, µ, C be nonnegativereal numbers such that

f ′′ + µf ≥ −δ, f(±T ) ≤ C, µT 2 < 2.

Then, for |s| ≤ T ,

f(s) ≤ C + T 2δ/2

1− µT 2/2.

Proof. First assume δ = C = 0. We must show that f ≤ 0. Assume, bycontradiction, that f has a positive maximum at s = s0. Replacing f by thefunction s 7→ f(−s), if necessary, we may assume that s0 ≥ 0. Hence

f(T ) = f(s0) +

∫ T

s0

(T − s)f ′′(s) ds

≥ f(s0)− µ

∫ T

s0

(T − s)f(s) ds

≥(1− µ

∫ T

s0

(T − s) ds

)f(s0)

≥ (1− µT 2/2)f(s0).

Thus f(T ) > 0, a contradiction. This proves the result in the case δ = C = 0.The general case can be reduced to the case δ = C = 0 by considering thefunction

g(s) := f(s) +δs2

2−(C +

δT 2

2

)1− µs2/2

1− µT 2/2.

It satisfies g′′ + µg ≥ 0 and g(±T ) ≤ 0. Hence g(s) ≤ 0 for |s| ≤ T and thisproves the lemma.

Proof of Proposition 4.2. Define f : R → R by

f(s) :=

∫ 1

0

∣∣v(ε2s, t)∣∣2 dt.

We prove that there is a constant µ = µ(V ) > 0 such that

f ′′ + µf + ε4 ≥ 0. (29)

17

To see this note that, by (7), we have

ε2∇s∇sv +∇t∇tv = ∇sv −∇t (∇Vt(u))

= ∇sv − (∇∂tu∇Vt) (u)−∇(∂tVt)(u)

and hence

(ε2∂s

2 + ∂t2) |v|2

2

= ε2 |∇sv|2 + |∇tv|2 +⟨ε2∇s∇sv +∇t∇tv, v

⟩

= ε2 |∇sv|2 + |∇tv|2 + 〈∇sv, v〉 − 〈(∇∂tu∇Vt)(u), v〉 − 〈∇(∂tVt)(u), v〉

≥ ε2

2|∇sv|2 + |∇tv|2 − ε−2 |v|

2

2− ‖∇∇V ‖∞ |∂tu| |v| − ‖∇∂tV ‖∞ |v|

=ε2

2|∇sv|2 + |∇tv|2 − ε−2 |v|

2

2

− ‖∇∇V ‖∞∣∣v − ε2∇sv

∣∣ |v| − ε2

2− ‖∇∂tV ‖2∞

ε−2 |v|22

≥ ε2

4|∇sv|2 + |∇tv|2 −

ε2

2

−(1 + 2ε2 ‖∇∇V ‖∞ + 2ε4 ‖∇∇V ‖2∞ + ‖∇∂tV ‖2∞

)ε−2 |v|

2

2.

This implies (ε4∂s

2 + ε2∂t2 + µ

)|v|2 ≥ −ε4

for a suitable constant µ = µ(V ) > 0. Integrating this inequality over theinterval 0 ≤ t ≤ 1 gives (29).

Next we prove that there is a constant C > 0 such that

∀s0 ∈ R∃s ∈ [s0, s0 + 1/√µ] such that f(s) ≤ C. (30)

This will follow from the energy and action estimates in the hypotheses of theproposition. More precisely, we have the following inequality for every s ∈ R:

c0 ≥ AV (us, vs) =

∫ 1

0

(〈v(s, t), ∂tu(s, t)〉 − |v(s, t)|2 /2− Vt(u(s, t))

)dt

=

∫ 1

0

(|v(s, t)|2 /2− ε2〈v(s, t),∇sv(s, t)〉 − Vt(u(s, t))

)dt

≥∫ 1

0

(|v(s, t)|2/4− ε4 |∇sv(s, t)|2 − Vt(u(s, t))

)dt.

Here we have used the fact that v = ∂tu+ ε2∇sv. This implies

f(s) ≤ 4c0 + 4 supS1×M

V +

∫ 1

0

ε4∣∣∇sv(ε

2s, t)∣∣2 dt

18

-r r0 |x|.

..

rεs

Pεr

2-r - rε.

.

...-r r

0 |x|.

..

s

Pr

2-r

...-r r

0 |x|.

.r−ε

s

P−εr

2-r + rε

...

Figure 1: Parabolic cylinders

for every s ∈ R. Now it follows from the energy estimate Eε(u, v) ≤ c0 that themean value of the function

s 7→∫ 1

0

ε4∣∣∇sv(ε

2s, t)∣∣2 dt

over any interval of length 1/õ is bounded above by c0

õ. Hence, for every

s0 ∈ R, there exists a point s ∈ R such that

s0 < s < s0 +1õ,

∫ 1

0

ε4∣∣∇sv(ε

2s, t)∣∣2 dt ≤ c0

õ.

This proves (30) with C = 4c0+4 supV + c0õ. Now it follows from (30) that,

for every s ∈ R, there exist points s1, s2 ∈ R such that

s− 1/√µ < s1 < s < s2 < s+ 1/

√µ, f(s1) ≤ C, f(s2) ≤ C.

Let s0 := (s1 + s2)/2 and T := (s2 − s1)/2. Then, by (29), the functions 7→ f(s0+s) satisfies the hypotheses of Lemma 4.3 on the interval [−T, T ] withthe above constants µ and C, and with δ = ε4. Hence

f(s) ≤ 2C + T 2ε4 ≤ 2C +ε4

µ

for every s ∈ R. This proves the proposition.

For n ∈ N denote by ∆n := ∂12 + · · ·+ ∂n

2 the standard Laplacian on Rn.Given positive real numbers r and ε let Br = Br(0) be the open ball of radius rin Rn and define the parabolic cylinders Pr, P

εr , P

−εr ⊂ Rn+1 by (see Figure 1)

P εr := (−r2 − εr, εr)×Br,

Pr := (−r2, 0)×Br,

P−εr := (−r2 + εr,−εr)×Br.

19

Lemma 4.4. For every n ∈ N there is a constant cn > 0 such that the followingholds for every r ∈ (0, 1]. If a ≥ 0 and w : R × Rn ⊃ Pr → R is C1 in thes-variable and C2 in the x-variable such that

(∆n − ∂s)w ≥ −aw, w ≥ 0,

then

w(0) ≤ cnear2

rn+2

∫

Pr

w.

Proof. For a = 0 this is a special case of a theorem by Gruber for parabolicdifferential operators with variable coefficients. (See Gruber [9, Theorem 2.1]with p = 1, θ = 1, λ = 1, σ = 1/2, R = r and f = 0; for an another proof seeLieberman [11, Theorem 7.21] with R = r, p = 1, ρ = 1/2, f = 0.)

To prove the result in general assume that w satisfies the hypotheses of thelemma and define f(s, x) := e−asw(s, x). Then

(∆n − ∂s)f = e−as(∆n − ∂s + a)w ≥ 0.

Hence, by Gruber’s theorem,

w(0) = f(0) ≤ cnrn+2

∫

Pr

f ≤ cnear2

rn+2

∫

Pr

w.

This proves the lemma.

Lemma 4.5. Let c2 be the constant in Lemma 4.4 with n = 2. Let ε > 0,r ∈ (0, 1], and a ≥ 0. If w : R×R ⊃ P ε

r → R is C1 in the s-variable and C2 inthe t-variable and satisfies

Lεw :=(ε2∂s

2 + ∂t2 − ∂s

)w ≥ −aw, w ≥ 0, (31)

then

w(0) ≤ 2c2ear2

r3

∫

P εr

w.

Proof. The idea of proof was suggested to us by Tom Ilmanen. Define a functionW on the domain Pr ⊂ R× R2 by

W (s, t, q) := w(s+ εq, t).

(Note that (s+ εq, t) ∈ P εr ⊂ R×R for every (s, t, q) ∈ Pr ⊂ R×R

2.) Then, byassumption, we have

(∆2 − ∂s)W (s, t, q) = (Lεw) (s+ εq, t) ≥ −aw(s+ εq, t) = −aW (s, t, q).

Hence it follows from Lemma 4.4 with n = 2 that

w(0) = W (0) ≤ c2ear2

r4

∫

Pr

W.

20

It remains to estimate the integral on the right hand side:

∫

Pr

W ≤∫ r

−r

∫ r

−r

∫ 0

−r2W (s, t, q) dsdqdt

=

∫ r

−r

∫ r

−r

∫ εq

−r2+εq

w(z, t) dzdqdt

≤∫ r

−r

∫ r

−r

∫ εr

−r2−εr

w(z, t) dzdqdt

= 2r

∫

P εr

w.

(32)

The first step uses the fact that W ≥ 0 and Br ⊂ [−r, r]× [−r, r]. The third stepuses the fact that w ≥ 0 and (−r2+εq, εq) ⊂ (−r2−εr, εr), since 0 ≤ q ≤ r.

Proof of Theorem 4.1. We prove that there is a constant a = a(V ) > 0 suchthat every solution (u, v) of (7) with 0 < ε ≤ 1 satisfies the inequality

(ε2∂s

2 + ∂t2 − ∂s

)|v|2 ≥ −a

(|v|2 + 1

). (33)

Namely, by (30), we have

(ε2∂s

2 + ∂t2 − ∂s

) |v|22

= ε2 |∇sv|2 + |∇tv|2 +⟨(ε2∇s∇s +∇t∇t −∇s

)v, v⟩

= ε2 |∇sv|2 + |∇tv|2 − 〈∇v−ε2∇sv∇Vt(u), v〉 − 〈∇(∂tVt)(u), v〉≥ ε2 |∇sv|2 + |∇tv|2 − ‖∇∇V ‖∞ |v|2 − ε2 ‖∇∇V ‖∞ |∇sv| |v| − ‖∇∂tV ‖∞ |v|

≥ ε2

2|∇sv|2 + |∇tv|2 −

(1 + 2 ‖∇∇V ‖∞ + ε2 ‖∇∇V ‖2∞

) |v|22

− ‖∇∂tV ‖2∞ .

This proves (33). Now let (s0, t0) ∈ R×S1 and apply Lemma 4.5 with r = 1 to

the function w : R× R ⊃ P ε1 → R, given by w(s, t) := |v(s+ s0, t+ t0)|2 + 1:

|v(s0, t0)|2 + 1 ≤ 2c2ea

∫ ε

−1−ε

∫ 1

−1

(|v(s+ s0, t+ t0)|2 + 1

)dtds

≤ 12c2ea

(1 + sup

s∈R

∫ 1

0

|v(s, t)|2 dt

).

Hence the result follows from Proposition 4.2.

5 Gradient bounds

Theorem 5.1. Let V ∈ C∞(S1 × M) and c0 > 0. Then there is a constantC > 0 such that the following holds. If 0 < ε ≤ 1 and (u, v) : R×S1 → TM is a

21

solution of (7) that satisfies (28), i.e. Eε(u, v) ≤ c0 and sups∈RAV (us, vs) ≤ c0,

then

|∂su(s, t)|2 + ε2 |∇sv(s, t)|2 ≤ CEε(u, v; [s− 1, s+ 1]× S1) ≤ Cc0. (34)

Here Eε(u, v;K) denotes the energy of (u, v) restricted to K ⊂ R× S1.

The proof of Theorem 5.1 has two steps. The first step is a bubbling argu-ment and establishes a weak form of the required L∞ estimate.

Lemma 5.2. Let V ∈ C∞(S1 ×M) and c0 > 0. Then the following holds.

(i) For every δ > 0 there is an ε0 > 0 such that every solution (u, v) : R×S1 →M of (7) and (28) with 0 < ε ≤ ε0 satisfies the inequality

ε2 ‖∂su‖∞ + ε3 ‖∇sv‖∞ ≤ δ. (35)

(ii) For every ε0 > 0 there is a constant c > 0 such that every solution (u, v) :R× S1 → M of (7) and (28) with ε0 ≤ ε ≤ 1 satisfies

‖∂su‖∞ + ε ‖∇sv‖∞ ≤ c.

The second step is a differential inequality for the energy density

e(s, t) :=1

2|∂su(s, t)|2 +

ε2

2|∇sv(s, t)|2 .

Lemma 5.3. Fix a potential V : S1 × M → R and a constant c0 > 0. Thenthere is a constant µ > 0 such that every solution (u, v) : R × S1 → M of (7)and (28) with 0 < ε ≤ 1 satisfies the differential inequality

Lεe :=(ε2∂s

2 + ∂t2 − ∂s

)e ≥ −µe.

Proof of Theorem 5.1. Let µ > 0 be the constant of Lemma 5.3 and c2 > 0 bethe constant of Lemma 4.5 with a = µ. Let (u, v) : R×S1 → TM be a solutionof (7) that satisfies (28). By Lemma 5.3, we may apply Lemma 4.5 with r = 1/2to the function w(s, t) := e(s0 + s, t0 + t). Hence

e(s0, t0) ≤ 16c2eµ/4

∫ s0+ε/2

s0−1/4−ε/2

∫ 1

0

e(s, t) dtds

≤ 8c2eµ/4

∫ s0+1

s0−1

∫ 1

0

(|∂su(s, t)|2 + ε2 |∇sv(s, t)|2

)dtds.

This proves (34) and the theorem.

For the proof of Lemma 5.2 we need the following lemma which is due toHofer. For proof see for example [14, Lemma 4.6.4].

22

Lemma 5.4. Let (X, d) be a complete metric space and f : X → R be anonnegative continuous function. Fix a point x ∈ X. Then there is a pointξ ∈ X and a positive number ρ ≤ 1 such that

d(x, ξ) < 2, supBρ(ξ)

f ≤ 2f(ξ), ρf(ξ) ≥ f(x).

Proof of Lemma 5.2. We prove (i). Suppose, by contradiction, that the resultis false. Then there are positive constants c0 and δ and a sequence of solutions(uν , vν) : R× S1 → M of (7) with ε = εν > 0 such that

Eεν (uν , vν) ≤ c0, sups∈R

AV (uν,s, vν,s) ≤ c0, limν→∞

εν = 0,

andε2ν ‖∂suν‖∞ + ε3ν ‖∇svν‖∞ > δ.

Hence there is a sequence zν = sν + itν ∈ C such that

ε2νcν ≥ δ, cν := |∂suν(sν , tν)|+ εν |∇svν(sν , tν)| . (36)

Modifying the sequence zν = sν + itν , if necessary, we may assume that sν = 0,0 ≤ tν ≤ 1, and

sup|z−zν |≤ρν

(|∂suν(z)|+ εν |∇svν(z)|) ≤ 2cν , ε2νρνcν ≥ δ, 0 < ρν ≤ 1. (37)

To see this, apply Lemma 5.4, to the metric space X = C and the functionf(z) := |∂suν(z)|+ εν |∇svν(z)|. Then apply a time shift to obtain sν = 0 anduse the periodicity in t to obtain 0 ≤ tν ≤ 1.

Now consider the sequence

wν = (uν , vν) : R× S1 → T ∗M

defined by

uν(s, t) := uν

(s

cν, tν +

t

ενcν

), vν(s, t) := ενgvν

(s

cν, tν +

t

ενcν

),

where g : TM → T ∗M denotes the isomorphism determined by the metric.This sequence satisfies the partial differential equation

∂suν − g−1∇tvν =1

cν∇Vtν+t/ενcν (uν), ∇svν + g∂tuν =

1

εν2cνvν .

We may write this equation in the form

∂J(wν)−∇Hν(wν ) = 0 (38)

where J ∈ J (T ∗M) is the almost complex structure determined by the metricand Hν : R× T ∗M → R is the Hamiltonian function

Hν(t, x, y) :=1

2εν2cν|y|2 + 1

cνVtν+t/ενcν (x).

23

By (37) we have1

ε2νcν≤ 1

δ.

Passing to a subsequence, if necessary, we may assume that the limits

t0 = limν→∞

tν , λ = limν→∞

1

ε2νcν

exist. It follows that the sequence Hν converges, uniformly with all derivativeson every compact subset of R × T ∗M , to the time independent Hamiltonianfunction

H(x, y) = limν→∞

Hν(t, x, y) =λ

2|y|2 .

Moreover, it follows from (36) and (37) that

|∂suν(0)|+ |∇svν(0)| = 1

andsup

|z|≤ρνcν

(|∂suν(z)|+ |∇svν(z)|) ≤ 2.

Moreover, by Theorem 4.1, there is a constant C > 0 such that |vν(s, t)| ≤ Cfor all s and t and hence

supν

ε−1ν ‖vν‖∞ ≤ C.

Here we used the uniform energy and action bounds. Hence it follows fromGromov’s graph trick (see [8] and [14]) and the standard elliptic bootstrappingtechnique for pseudoholomorphic curves (see [13, Appendix B]) that there is asubsequence, still denoted by wν = (uν , vν), that converges, uniformly with allderivatives on every compact subset of C, to a solution w = (u, v) : C → T ∗Mof the differential equation,

∂J(w) = ∇H(w).

Moreover, v(s, t) ≡ 0 and hence u(s, t) is constant. But this contradicts the factthat

|∂su(0)|+ |∇sv(0)| = limν→∞

(|∂suν(0)|+ |∇svν(0)|) = 1.

Thus we have proved assertion (i). The proof of (ii) is almost word by wordthe same, except that εν no longer converges to zero and instead cν diverges toinfinity. So the limit Hamiltonian is H = 0 and so, by removal of singularities,one obtains in the limit a nonconstant J-holomorphic sphere w : S2 → T ∗M ,which cannot exist since the symplectic form on T ∗M is exact. This proves thelemma.

Proof of Lemma 5.3. Straightforward calculation shows that

Lεe = |∇t∂su|2 + ε2 |∇s∂su|2 + ε2 |∇t∇sv|2 + ε4 |∇s∇sv|2 + U + ε2V, (39)

24

where

U :=⟨∂su,

(ε2∇s∇s +∇t∇t −∇s

)∂su⟩,

V :=⟨∇sv,

(ε2∇s∇s +∇t∇t −∇s

)∇sv⟩.

We shall prove that U and V satisfy the pointwise inequality

|U |+ ε2|V | ≤ µe+1

2

(|∇t∂su|2 + ε2 |∇s∂su|2 + ε2 |∇t∇sv|2 + ε4 |∇s∇sv|2

)(40)

for a suitable constant µ > 0. The required estimate Lεe ≥ −µe follows imme-diately from (39) and (40). To prove (40) we observe that, by (7),

(ε2∇s∇s +∇t∇t −∇s

)∂su

= ε2∇s∇s (∇tv +∇Vt(u)) +∇t∇s

(v − ε2∇sv

)−∇s (∇tv +∇Vt(u))

= ε2 [∇s∇s,∇t] v + [∇t,∇s] v −∇s∇Vt(u) + ε2∇s∇s∇Vt(u)

= 2ε2R(∂su, ∂tu)∇sv + ε2 (∇∂suR) (∂su, ∂tu)v −R(∂su, ∂tu)v

+ ε2R(∇s∂su, ∂tu)v + ε2R(∂su,∇s∂tu)v −∇∂su∇Vt(u)

+ ε2∇2∇Vt(u)(∂su, ∂su) + ε2∇∇s∂su∇Vt(u).

(41)

Now fix a sufficiently small constant δ > 0 and choose ε0 > 0 such that theassertion of Lemma 5.2 (i) holds. Choose C > 0 such that the assertion ofTheorem 4.1 holds and assume 0 < ε ≤ ε0 ≤ δ/C. Then, by Theorem 4.1 andLemma 5.2, we have

ε2 ‖∂su‖∞ ≤ δ, ε3 ‖∇sv‖∞ ≤ δ, ‖v‖∞ ≤ C, ε ‖∂tu‖∞ ≤ 2δ. (42)

The last estimate uses the identity ∂tu = v − ε2∇sv. Differentiate the latterequation covariantly with respect to s to obtain

∇sv = ∇s∂tu+ ε2∇s∇sv, ∂tu = v − ε2∇t∂su− ε4∇s∇sv. (43)

Now take the pointwise inner product of (41) with ∂su and estimate the resultingeight expressions separately. The inequalities (42) show that the last five termsare bounded by the right hand side of (40). For the first three terms we argueas follows. Half the first term can be expressed in the form

ε2⟨∂su,R(∂su, ∂tu)∇sv

⟩

= ε2⟨∂su,R(∂su, v)∇t∂su

⟩+ ε4

⟨∂su,R(∂su, v)∇s∇sv

⟩

− ε4⟨∂su,R(∂su,∇t∂su)∇t∂su

⟩− ε6

⟨∂su,R(∂su,∇t∂su)∇s∇sv

⟩

− ε6⟨∂su,R(∂su,∇s∇sv)∇t∂su

⟩− ε8

⟨∂su,R(∂su,∇s∇sv)∇s∇sv

⟩.

Here we have replaced ∂tu and ∇sv by the expressions in (43). In the first twoterms we eliminate one of the factors ∂su by using the inequality ε2|∂su| ≤ δand in the last four terms we eliminate both factors ∂su by the same inequality.The next two terms in our expression for U have the form

ε2〈∂su, (∇∂suR) (∂su, ∂tu)v〉 − 〈∂su,R(∂su, ∂tu)v〉.

25

Replace ∂tu by the expression in (43) and elimate in each of the resulting sum-mands one or two of the factors ε2∂su as above. This proves the requiredestimate for U and 0 < ε ≤ ε0.

To estimate V we observe that, by (7),

(ε2∇s∇s +∇t∇t −∇s

)∇sv

= ∇s∇s(ε2∇sv − v) +∇t∇s∇tv +∇t([∇t,∇s]v)

= −∇s∇t∂su+∇t∇s(∂su−∇Vt(u))−∇t(R(∂su, ∂tu)v)

= −R(∂su, ∂tu)∂su+R(∂su, ∂tu)∇Vt(u)−∇t(R(∂su, ∂tu)v)

−∇s∇t∇Vt(u)

= −2R(∂su, ∂tu)∂su+ 2R(∂su, ∂tu)∇Vt(u)

− (∇∂tuR) (∂su, ∂tu)v −R(∇t∂su, ∂tu)v −R(∂su,∇t∂tu)v

−∇2∇Vt(u)(∂tu, ∂su)−∇∇t∂su∇Vt(u)−∇∂su(∇∂tVt)(u).

(44)

The last step uses the identity ∇tv = ∂su − ∇Vt(u). Now take the pointwiseinner product with ε2∇sv. Then the first term has the same form as the onedicussed above. In the second, fourth, and sixth term we estimate ε|∂tu| by 2δ.The last two terms are obviously bounded by the right hand side of (40). Thisleaves the terms three and five. In the third term we estimate ε2|∂tu|2 by 4δ2

and use the identity ∇sv = ∇s∂tu + ε2∇s∇sv of (43). For term five we use theidentity

∇t∂tu = ∇t(v − ε2∇sv) = ∂su−∇Vt(u)− ε2∇t∇sv

to obtain the expression

ε2〈∇sv,R(∂su, ∂su−∇Vt(u)− ε2∇t∇sv)v〉= −ε2〈∇sv,R(∂su,∇Vt(u))v〉 − ε4〈∇sv,R(∂su,∇t∇sv)v〉.

In the last summand we use the estimate ε2|∂su| ≤ δ. This proves (40) for0 < ε ≤ ε0. For ε0 ≤ ε ≤ 1 the estimate (40) follows immediately from (41),(44), and Lemma 5.2 (ii).

6 Estimates of the second derivatives

Theorem 6.1. Fix a potential V : S1 ×M → R and a constant c0 > 0. Thenthere is a constant c > 0 such that the following holds. If 0 < ε ≤ 1 and(u, v) : R× S1 → TM is a solution of (7) that satisfies (28) then

ε1/2−1/p ‖∇t∂su‖Lp([−T,T ]×S1) + ε ‖∇s∂su‖Lp([−T,T ]×S1)

+ ε1−1/p ‖∇t∇sv‖Lp([−T,T ]×S1) + ε3/2 ‖∇s∇sv‖Lp([−T,T ]×S1)

≤ c√Eε(u, v; [−T − 200/ε, T + 200/ε]× S1)

for all T > 0 and 2 ≤ p ≤ ∞.

26

Proposition 6.2. Fix a potential V : S1 × M → R and a constant c0 > 0.Then there is a constant c > 0 such that the following holds for every ε ∈ (0, 1]and every solution (u, v) : R× S1 → TM of (7) that satisfies (28). Denote

2u0 := |∂su|2 + ε |∇sv|2 ,2v0 := |∇t∂su|2 + ε2 |∇s∂su|2 + ε |∇t∇sv|2 + ε3 |∇s∇sv|2 ,2u1 := |∇t∂su|2 + ε |∇s∂su|2 + ε |∇t∇sv|2 + ε2 |∇s∇sv|2

and Lε := ε2∂s∂s + ∂t∂t − ∂s. Then

Lεu0 ≥ v0 − c|∂su|2 − cε2 |∇sv|2 , (45)

Lε

(u0 + u1

)≥ −c

(u0 + u1

). (46)

We show first how Theorem 6.1 follows from Proposition 6.2. This requiresLemma 6.4 below the proof of which, in turn, relies on the next lemma. Letn be a positive integer and r, R be positive real numbers. Recall the definitionof the parabolic domain PR ⊂ R × Rn in Section 4. The elements of PR aredenoted by (s, x) = (s, x1, . . . , xn) and

∆ = ∂12 + · · · ∂n2

denotes the Laplace operator.

Lemma 6.3. Let u : R × Rn ⊃ PR+r → R be C1 in the s-variable and C2 inthe x-variable and f, g : PR+r → R be continous functions such that

(∆u− ∂s)u ≥ f − g, u ≥ 0, f ≥ 0, g ≥ 0.

Then ∫

PR

f ≤∫

PR+r

g +

(4

r2+

1

Rr

)∫

PR+r\PR

u.

Proof. The proof rests on the following two inequalities. Let Br ⊂ Rn bethe open ball of radius r centered at zero. Then, for every smooth functionu : Rn → [0,∞), we have

∫

∂Br

∂u

∂ν= −n− 1

r

∫

∂Br

u+d

dr

∫

∂Br

u ≤ d

dr

∫

∂Br

u (47)

(see [10, Theorem 2.1]). Secondly, every smooth function u : R × Rn → [0,∞)satisfies

d

dσ

∫ 0

−(R+σ)2

(∫

∂BR+σ

u(s, ·))ds =

∫ 0

−(R+σ)2

(d

dσ

∫

∂BR+σ

u(s, ·))ds

+ 2(R+ σ)

∫

∂BR+σ

u(−(R+ σ)2, ·)

≥∫ 0

−(R+σ)2

(d

dσ

∫

∂BR+σ

u(s, ·))ds.

(48)

27

Now suppose u, f , g satisfy the assumptions of the lemma. Then, for 0 ≤ σ ≤ r,

∫

PR

f −∫

PR+r

g

≤∫

PR+σ

(∆u − ∂su)

=

∫ 0

−(R+σ)2

(∫

∂BR+σ

∂u

∂ν(s, ·)

)ds−

∫

BR+σ

(u(0, ·)− u(−(R+ σ)2, ·)

)dx

≤∫ 0

−(R+σ)2

(d

dσ

∫

∂BR+σ

u(s, ·))ds+

∫

BR+σ

u(−(R+ σ)2, x) dx

≤ d

dσ

∫ 0

−(R+σ)2

(∫

∂BR+σ

u(s, ·))ds+

∫

BR+σ

u(−(R+ σ)2, x) dx.

Here the first step uses the inclusions PR ⊂ PR+σ ⊂ PR+r. The third stepfollows from (47) and the last from (48). Now integrate this inequality over theinterval 0 ≤ σ ≤ t, with r/2 ≤ t ≤ r, to obtain

r

2

(∫

PR

f −∫

PR+r

g

)

≤∫ 0

−(R+t)2

(∫

∂BR+t

u(s, ·))ds+

∫ r

0

(∫

BR+σ

u(−(R+ σ)2, ·))dσ

≤∫ 0

−(R+r)2

(∫

∂BR+t

u(s, ·))ds+

1

2R

∫ −R2

−(R+r)2

∫

BR+r

u(s, x) dxds.

Here the last step follows by substituting s = −(R + σ)2 and using√−s ≥ R.

Integrate this inequality again over the interval r/2 ≤ t ≤ r to obtain

r

2

(∫

PR

f −∫

PR+r

g

)≤ 2

r

∫

PR+r\PR

u+1

2R

∫

PR+r\PR

u.

This proves Lemma 6.3.

Lemma 6.4. Let ε,R, r be positive real numbers. Let u : R2 ⊃ P εR+r → R be a

C2 function and f, g : P εR+r → R be continous functions such that

(ε2∂s

2 + ∂t2 − ∂s

)u ≥ f − g, u ≥ 0, f ≥ 0, g ≥ 0.

Then∫

P−εR/2

f ≤ 2(R+ r)

R

∫

P εR+r

g +2(R+ r)

R

(4

r2+

1

Rr

)∫

P εR+r

u.

28

Proof. The idea of proof is as in Lemma 4.5. Increase the dimension of thedomain from two to three and apply Lemma 6.3 with n = 2. Define functionsU, F,G on PR+r ⊂ R× R2 by

U(s, t, q) := u(s+ εq, t), F (s, t, q) := f(s+ εq, t), G(s, t, q) := g(s+ εq, t).

The new variable σ := s+ εq satisfies (σ, t) ∈ P εR+r ⊂ R×R whenever (s, t, q) ∈

PR+r ⊂ R×R2. Use the differential inequality in the assumption of the lemmato conclude (∆− ∂s)U ≥ F − G, where ∆ := ∂2

t + ∂2q . Thus Lemma 6.3 with

n = 2 yields

∫

PR

F ≤∫

PR+r

G+

(4

r2+

1

Rr

)∫

PR+r

U

≤ 2(R+ r)

∫

P εR+r

g + 2(R+ r)

(4

r2+

1

Rr

)∫

P εR+r

u.

The last step uses (32). By definition of F

∫

PR

F =

∫

BR⊂R2

(∫ 0

−R2

f(s+ εq, t) ds

)dqdt

≥∫ R/2

−R/2

∫ R/2

−R/2

∫ εq

−R2+εq

f(σ, t) dσdqdt

≥∫ R/2

−R/2

∫ R/2

−R/2

∫ −εR/2

−R2+εR/2

f(σ, t) dσdqdt

= R

∫ R/2

−R/2

∫ −εR/2

−R2+εR/2

f(σ, t) dσdt

≥ R

∫

P−εR/2

f.

This proves the lemma.

Proof of Theorem 6.1. We first prove the estimate for p = 2. By (45) in Propo-sition 6.2 and Lemma 6.4, we have

∫

P−εR/2

v0 ≤ 2(R+ r)

R

∫

P εR+r

c(|∂su|2 + ε2 |∇sv|2

)

+2(R+ r)

R

(4

r2+

1

Rr

)∫

P εR+r

u0.

Choose R = r = 6/√ε. Then

P−εR/2 ⊃ [−9/ε+ 3,−3]× [−3/

√ε, 3/

√ε],

P εR+r ⊂ [−144/ε− 12, 12]× [−12/

√ε, 12/

√ε].

29

Moreover, 2(R+r)/R = 4 and 4/r2+1/Rr ≤ ε. Since εu0 = ε |∂su|2+ε2 |∇sv|2,it follows that there is a constant c1 > 0 such that, for every s ∈ R, we have

‖∇t∂su‖2L2([s−9/ε+3,s−3]×S1) + ε2 ‖∇s∂su‖2L2([s−9/ε+3,s−3]×S1)

+ ε ‖∇t∇sv‖2L2([s−9/ε+3,s−3]×S1) + ε3 ‖∇s∇sv‖2L2([s−9/ε+3,s−3]×S1)

≤ c1Eε(u, v; [s− 144/ε− 12, s+ 12]× S1).

Here we use the fact that the interval [−3/√ε, 3/

√ε] covers the circle at least

k times and the interval [−12/√ε, 12/

√ε] covers the circle at most 5k times,

where k is the positive integer k := [6/√ε] ≥ 5/

√ε.

Now cover the interval [−T, T ] by ℓ adjacent intervals of the form

Ij := [sj − 9/ε− 3, sj + 3],

where ℓ− 1 = [2T/(9/ε− 6)]. Then the intervals

Kj := [sj − 144/ε− 12, sj + 12]

are contained in [−T − 135/ε− 15, T + 9/ε+ 9]. Note that sj+1 − sj = 9/ε− 6and

0 < ε ≤ 1 =⇒ |Kj||Ij |

=144/ε+ 24

9/ε− 6≤ 168

3= 56.

Hence at most 56 succesive intervals of the Kj have nonempty intersection. Thisimplies

‖∇t∂su‖2L2([−T,T ]×S1) + ε2 ‖∇s∂su‖2L2([−T,T ]×S1)

+ ε ‖∇t∇sv‖2L2([−T,T ]×S1) + ε3 ‖∇s∇sv‖2L2([−T,T ]×S1)

≤ 56c1Eε(u, v; [−T − 135/ε− 15, T + 9/ε+ 9]× S1).

(49)

Thus we have proved the L2 estimate.Next we consider the case p = ∞. By (46) in Proposition 6.2, we may apply

Lemma 4.5, with a = c and r = 1, to the function w = u0 + u1 to obtain

(u0 + u1) (s0, t0) ≤2ec

c2

∫ s0+ε

s0−1−ε

∫ t0+1

t0−1

(u0 + u1) (s, t) dtds

≤ c3ε−1Eε(u, v; [s0 − 135/ε− 17, s0 + 9/ε+ 11]× S1).

The second inequality uses (49) with T = 2. The desired L∞ estimate nowfollows by taking the maximum over all s0 ∈ [−T, T ].

To prove the result for general p we apply the interpolation inequality

‖ξ‖p ≤ ‖ξ‖1−2/p∞ ‖ξ‖2/p2

to the terms on the left hand side of the estimate and use the results for p = 2and p = ∞. This proves the theorem.

30

Proof of Proposition 6.2. By the apriori estimate for v in Theorem 4.1 and thegradient estimate for ∂su and ∇sv in Theorem 5.1, there is a constant c1 =c1(V, c0) > 0 such that

‖∂tu‖∞ + ‖v‖∞ + ‖∂su‖∞ + ε ‖∇sv‖∞ ≤ c1 (50)

for every solution (u, v) of (7) that satisfies (28).To prove (45) we observe that

Lεu0 = 2v0 + U + εV,

U =⟨(ε2∇s∇s +∇t∇t −∇s

)∂su, ∂su

⟩

V =⟨(ε2∇s∇s +∇t∇t −∇s

)∇sv,∇sv

⟩.

It follows from (41), (44), and (50) that

|U |+ ε|V | ≤ v0 + c2

(|∂su|2 + ε2 |∇sv|2

)

=1

2

(|∇t∂su|2 + ε2 |∇s∂su|2 + ε |∇t∇sv|2 + ε3 |∇s∇sv|2

)

+ c2

(|∂su|2 + ε2 |∇sv|2

).

The only term that requires attention is the factor∇t∂tu in one of the summandsin (44). For this term we can use the identity ∇t∂tu = ∂su−∇Vt(u)− ε2∇t∇svas in the proof of Lemma 5.3. This proves (45).

The proof of (46) will follow from the inequality

Lεu1 ≥ v1 − c3 |∂su|2 − c3u1, (51)

2v1 := |∇t∇t∂su|2 + ε2 |∇s∇t∂su|2 + ε |∇t∇t∇sv|2 + ε3 |∇t∇s∇sv|2

+ ε |∇t∇s∂su|2 + ε3 |∇s∇s∂su|2 + ε2 |∇t∇s∇sv|2 + ε4 |∇s∇s∇sv|2 .To see this simply add the inequalities (45) and (51).

To prove (51) we observe that

Lεu1 = 2v1 + f1 (52)

where

f1 = 〈∇t∂su, I〉+ ε 〈∇s∂su, II〉+ ε 〈∇t∇sv, III〉+ ε2 〈∇s∇sv, IV 〉

and

I :=(ε2∇s∇s +∇t∇t −∇s

)∇t∂su,

II :=(ε2∇s∇s +∇t∇t −∇s

)∇s∂su,

III :=(ε2∇s∇s +∇t∇t −∇s

)∇t∇sv,

IV :=(ε2∇s∇s +∇t∇t −∇s

)∇s∇sv.

31

We write II and III in the form

II = ∇s

(ε2∇s∇s +∇t∇t −∇s

)∂su+ [∇t∇t,∇s]∂su,

III = ∇t

(ε2∇s∇s +∇t∇t −∇s

)∇sv − [∇s,∇t]∇sv + ε2[∇s∇s,∇t]∇sv.

The last term in II and the last two terms in III can be estimated easily in therequired form (after taking the inner product with ε∇s∂su, respectively ε∇t∇sv,and inserting the relevant curvature terms for the commutators). The importantterms are the first in each expression. We insert the formula (41) in II and (44)into III and carry out covariant differentiation with respect to s in the firstcase and with respect to t in the second case. Then the only difficult terms arethose that lead to cubic terms in the second derivatives. After taking the innerproduct with ε∇s∂su, respectively ε∇t∇sv, they have the form

II0 := 2ε3〈R(∇s∂su,∇s∂tu)v,∇s∂su〉,III0 := 2ε〈R(∇t∂tu,∇t∂su)v,∇t∇sv〉.

Now insert the expressions

∇s∂su = ∇s (∇tv +∇Vt(u)) , ∇t∂tu = ∇t

(v − ε2∇sv

)

into II0 and III0, respectively. Then the only difficult remaining terms arethe ones involving again three second derivatives. After replacing ∇s∇tv by∇t∇sv +R(∂su, ∂tu)v we obtain

II1 := 2ε3〈R(∇t∇sv,∇s∂tu)v,∇s∂su〉,III1 := −2ε3〈R(∇t∇sv,∇t∂su)v,∇t∇sv〉.

The sum of these terms is

II1 + III1 = 2ε3〈R(∇t∇sv,∇s∂tu)v,∇s∂su−∇t∇sv〉= 2ε3〈R(∇t∇sv,∇s∂tu)v,∇s(∂su−∇tv) +R(∂su, ∂tu)v〉= 2ε3〈R(∇t∇sv,∇s∂tu)v,∇∂suVt(u) +R(∂su, ∂tu)v〉

and can be estimated in the required fashion. The terms I and IV can be treatedsimilarly. Inspecting (41) and (44) and noting that the s- and t-derivatives areinterchanged we find that the difficult cubic terms in the second derivatives donot appear in I and IV . This proves the inequality (51) and the proposition.

7 Uniform exponential decay

Theorem 7.1. Suppose SV is Morse and let a ∈ R be a regular value of SV .Then there exist positive constants δ, c, ρ such that the following holds. If x± ∈Pa(V ), 0 < ε ≤ 1, T0 > 0, and (u, v) ∈ Mε(x−, x+;V ) satisfies

Eε(u, v; (R \ [−T0, T0])× S1) < δ, (53)

32

then

Eε(u, v; (R \ [−T, T ])× S1) ≤ ce−ρ(T−T0)Eε(u, v; (R \ [−T0, T0])× S1)

for every T ≥ T0 + 1.

Corollary 7.2. Suppose SV is Morse and let x± ∈ P(V ). Then there existpositive constants δ, c, ρ such that the following holds. If 0 < ε ≤ 1, T0 > 0, and(u, v) ∈ Mε(x−, x+;V ) satisfies (53) then

|∂su(s, t)|2 + ε2 |∇sv(s, t)|2 ≤ ce−ρ|s|Eε(u, v; (R \ [−T0, T0])× S1) (54)

for every |s| ≥ T0 + 2.

Proof. Theorem 5.1 and Theorem 7.1.

The proof of Theorem 7.1 is based on the following lemma and proposition.

Lemma 7.3 (The Hessian). Suppose SV is Morse and fix a ∈ R. Then thereare positive constants δ0 and c such that the following is true. If x0 ∈ Pa(V )and (x, y) ∈ C∞(S1, TM) satisfy

x = expx0(ξ0), y = Φ(x0, ξ0)(∂tx0 + η0), ‖ξ0‖W 1,2 + ‖η0‖∞ ≤ δ0,

then

‖ξ‖2 + ‖∇tξ‖2 + ‖η‖2 + ‖∇tη‖2≤ c

(‖∇tη +R(ξ, ∂tx)y +∇ξ∇Vt(x)‖2 + ‖∇tξ − η‖2

)

for all ξ, η ∈ Ω0(S1, x∗TM).

Proof. The operator

Hx,y(ξ, η) := (−∇tη −R(ξ, ∂tx)y −∇ξ∇Vt(x),∇tξ − η)

on L2(S1, x∗TM ⊕x∗TM) with dense domain W 1,2(S1, x∗TM ⊕x∗TM) is self-adjoint if y = ∂tx. In the case (x, y) = (x0, ∂tx0) it is bijective, because SV isMorse. Hence the result is a consequence of the open mapping theorem. Sincebijectivity is preserved under small perturbations (with respect to the operatornorm), the result for general pairs (x, y) follows from continuous dependence ofthe operator family on the pair (x, y) with respect to the W 1,2-topology on xand the L∞-topology on y. The set Pa(V ) is finite, because SV is Morse (see[21]). Hence we may choose the same constants δ0 and c for all x0 ∈ Pa(V ).

Lemma 7.4. Let SV be Morse and a ∈ R be a regular value of SV . Then,for every δ0 > 0, there is a constant δ1 > 0 such that the following is true. If(x, y) : S1 → TM is a smooth loop such that

AV (x, gy) ≤ a, ‖∇ty +∇Vt(x)‖∞ + ‖∂tx− y‖∞ < δ1,

33

then there is a periodic orbit x0 ∈ Pa(V ) and a pair of vector fields ξ0, η0 ∈Ω0(S1, x0

∗TM) such that

x = expx0(ξ0), y = Φ(x0, ξ0)(∂tx0 + η0),

and‖ξ0‖∞ + ‖∇tξ0‖∞ + ‖η0‖∞ + ‖∇tη0‖∞ ≤ δ0.

Proof. First note that

1

2

∫ 1

0

|y(t)|2 = AV (x, gy) +

∫ 1

0

(Vt(x(t)) − 〈y(t), x(t)− y(t)〉

)dt

≤ a+ ‖V ‖∞ +

∫ 1

0

(1

4|y(t)|2 + |x(t)− y(t)|2

)dt.

Hence, assuming δ1 ≤ 1, we have

‖y‖22 ≤ 4 (a+ ‖V ‖∞ + 1) .

Now∣∣∣∣d

dt|y|2∣∣∣∣ =

∣∣2〈y,∇ty +∇Vt(x)〉 − 2〈y,∇Vt(x)〉∣∣

≤ 2 (δ1 + ‖∇Vt‖∞) |y|≤ (‖∇Vt‖∞ + 1)

2+ |y|2 .

Integrate this inequality to obtain

|y(t1)|2 − |y(t0)|2 ≤ (‖∇Vt‖∞ + 1)2+ ‖y‖22

for t0, t1 ∈ [0, 1]. Integrating again over the interval 0 ≤ t0 ≤ 1 gives

‖y‖∞ ≤√(‖∇Vt‖∞ + 1)2 + 2 ‖y‖22 ≤ c (55)

where c2 := (‖∇Vt‖∞ + 1)2+ 8 (a+ ‖V ‖∞ + 1).

Now suppose that the assertion is wrong. Then there is a δ0 > 0 and asequence of smooth loops (xν , yν) : S

1 → TM satisfying

AV (xν , gyν) ≤ a, limν→∞

(‖∇tyν +∇Vt(xν)‖∞ + ‖∂txν − yν‖∞

)= 0,

but not the conclusion of the lemma for the given constant δ0. By (55), wehave supν ‖yν‖∞ < ∞. Hence supν ‖∂txν‖∞ < ∞ and also supν ‖∇tyν‖∞ < ∞.Hence, by the Arzela–Ascoli theorem, there exists a subsequence, still denotedby (xν , yν), that converges in the C0-topology. Our assumptions guarantee thatthis subsequence actually converges in the C1-topology. Let (x0, y0) : S

1 → TMbe the limit. Then ∂tx0 = y0 and ∇ty0 + ∇Vt(x0) = 0. Hence x0 ∈ Pa(V )and (xν , yν) converges to (x0, ∂tx0) in the C1-topology. This contradicts ourassumption on the sequence (xν , yν) and hence proves the lemma.

34

Proof of Theorem 7.1. To begin with note that

SV (x) ≥ −‖V ‖∞

for every x ∈ P(V ). Hence, with

c0 := a+ ‖V ‖∞ ,

we haveSV (x

−) ≤ c0, SV (x−)− SV (x

+) ≤ c0

for all x± ∈ Pa(V ). Let C > 0 be the constant of Theorem 5.1 with this choiceof c0. Let δ0 and c be the constants of Lemma 7.3 and δ1 > 0 the constant ofLemma 7.4 associated to a and δ0. Then choose δ > 0 such that

√Cδ ≤ δ1.

In the remainder of the proof we will sometimes use the notation us(t) :=u(s, t) and vs(t) := v(s, t). Moreover, ‖·‖ will always denote the L2 norm on S1

and ‖·‖∞ the L∞ norm on S1.Now let x± ∈ Pa(V ), 0 < ε ≤ 1, and T0 > 0, and suppose (u, v) ∈

Mε(x−, x+;V ) satisfies (53). Then, by Theorem 5.1, we have

‖∂sus‖∞ + ε ‖∇svs‖∞ ≤√CEε(u, v; [s− 1, s+ 1]× S1) ≤

√Cδ ≤ δ1 (56)

for |s| ≥ T0 + 1. Hence, by Lemma 7.4, we know that, for every s ∈ R with|s| ≥ T0 + 1, there is a periodic orbit xs ∈ Pa(V ) such that the C1-distancebetween (us, vs) and (xs, ∂txs) is bounded by δ0. Hence we can apply Lemma 7.3to the pair (us, vs) and the vector fields (∂sus,∇svs) for |s| ≥ T0 + 1. Since

∇t∇sv +R(∂su, ∂tu)v +∇∂su∇Vt(u) = ∇s∂su

and∇t∂su−∇sv = −ε2∇s∇sv,

we obtain from Lemma 7.3 that

‖∂sus‖22 + ‖∇t∂sus‖22 + ‖∇svs‖22 + ‖∇t∇svs‖22≤ c

(‖∇s∂sus‖22 + ε4 ‖∇s∇svs‖22

).

(57)

for |s| ≥ T0 + 1.Define the function f : R → [0,∞) by

f(s) :=1

2

∫ 1

0

(|∂su(s, t)|2 + ε2 |∇sv(s, t)|2

)dt.

We shall prove that

f ′′(s) ≥ 1

cf(s) (58)

for |s| ≥ T0 + 1.

35

Fix a real number s with |s| ≥ T0 + 1. In the next identity all norms andinner products are understood in L2(S1, u∗

sTM) and we drop the subscript sfor us and vs. Then

f ′′(s) = ‖∇s∂su‖2 + ε2 ‖∇s∇sv‖2 + 〈∂su,∇s∇s∂su〉+ ε2 〈∇sv,∇s∇s∇sv〉= ‖∇s∂su‖2 + ε2 ‖∇s∇sv‖2

+ 〈∂su,∇s∇s(∇tv +∇Vt(u))〉+ 〈∇sv,∇s∇s(v − ∂tu)〉= ‖∇s∂su‖2 + ε2 ‖∇s∇sv‖2

+ 〈∂su, [∇s∇s,∇t]v +∇t∇s∇sv +∇s∇s∇Vt(u)〉+ 〈∇sv,∇s∇sv − [∇s,∇t]∂su−∇t∇s∂su〉

= ‖∇s∂su‖2 + ε2 ‖∇s∇sv‖2

+ 〈∂su,∇s[∇s,∇t]v + [∇s,∇t]∇sv +∇s∇s∇Vt(u)〉− 〈∇s∂tu,∇s∇svs〉+ 〈∇sv,∇s∇sv − [∇s,∇t]∂su〉+ 〈∇t∇sv,∇s∂su〉

= ‖∇s∂su‖2 + ε2 ‖∇s∇sv‖2

+ 〈∂su,∇s[∇s,∇t]v + [∇s,∇t]∇sv +∇s∇s∇Vt(u)〉−⟨∇s(v − ε2∇sv),∇s∇sv

⟩+ 〈∇sv,∇s∇sv − [∇s,∇t]∂su〉

+ 〈[∇t,∇s]v +∇s(∂su−∇Vt(u)),∇s∂su〉= 2 ‖∇s∂su‖2 + 2ε2 ‖∇s∇sv‖2 + 〈∂su,∇s∇s∇Vt(u)〉 − 〈∇s∇Vt(u),∇s∂su〉+ 〈∂su, 3R(∂su, ∂tu)∇sv〉+ 〈∂su, (∇∂suR)(∂su, ∂tu)v〉+ 〈∂su,R(∂su,∇s∂tu)v〉+ 〈∂su,R(∇s∂su, ∂tu)v〉 − 〈R(∂su, ∂tu)v,∇s∂su〉

= 2 ‖∇s∂su‖2 + 2ε2 ‖∇s∇sv‖2 +⟨∂su,∇2∇Vt(u)(∂su, ∂su)

⟩

+ 〈∂su, 3R(∂su, ∂tu)∇sv〉+ 〈∂su, (∇∂suR)(∂su, ∂tu)v〉+ 〈∂su,R(∂su,∇sv)v〉 − ε2 〈∂su,R(∂su,∇s∇sv)v〉− 〈∂su,R(∂tu− v, v)∇s∂su〉 .

The last identity follows from the pointwise symmetry of the covariant Hessianof V , the formula ∂tu = v − ε2∇sv, and first Bianchi identity. Since the L∞

norms of ∂tu and v are uniformly bounded, there is a constant c′ > 0 such that

f ′′(s) ≥ 2 ‖∇s∂sus‖22 + 2ε2 ‖∇s∇svs‖22

− c′ ‖∂sus‖∞(‖∂sus‖22 + ‖∂sus‖2 ‖∇svs‖2

)

− c′ε2 ‖∂sus‖∞(‖∂sus‖2 ‖∇s∇sv‖2 + ‖∇svs‖2 ‖∇s∂sus‖2

)

≥ ‖∇s∂sus‖22 + ε2 ‖∇s∇svs‖22 .

To understand the last step note that, by (56), we have ‖∂sus‖∞ ≤√Cδ and so

the inequality follows from (57), provided that δ > 0 is sufficiently small. Nowuse (57) again to obtain (58).

36

Thus we have proved that f ′′(s) ≥ ρ2f(s) for |s| ≥ T0 + 1, where ρ :=c−1/2. Since f(s) does not diverge to infinity as |s| → ∞ it follows by standardarguments (see for example [4, 16]) that f(s) ≤ e−ρ(s−T0−1)f(T0 + 1) for s ≥T0 + 1 and similarly for s ≤ −T0 − 1. This proves the theorem.

8 Time shift

The next theorem establishes local surjectivity for the map T ε constructed inTheorem 3.1. The idea is to prove that, after a suitable time shift, the pairζ = (ξ, η) with uε = expu(ξ) and vε = Φ(u, ξ)(∂tu + η) satisfies the hypothesisζ ∈ im (Dε

u)∗ of Theorem 3.3. The neighbourhood, in which the next theorem

establishes surjectivity, depends on ε.

Theorem 8.1. Assume SV is Morse–Smale and fix a regular value a ∈ R ofSV . Fix two constants C > 0 and p > 1. Then there are positive constants δ, ε0,and c such that ε0 ≤ 1 and the following holds. If x± ∈ Pa(V ) is a pair of indexdifference one, u ∈ M0(x−, x+;V ), 0 < ε ≤ ε0, (u

ε, vε) ∈ Mε(x−, x+;V ), anduε = expu(ξ

ε), where ξε ∈ Ω0(R× S1, u∗TM) satisfies

‖ξε‖∞ ≤ δε1/2, ‖ξε‖p ≤ δε1/2, ‖∇tξε‖p ≤ C, (59)

then there is a real number σ such that

(uε, vε) = T ε(u(σ + ·, ·)), |σ| < c(‖ξε‖p + ε2).

Proof. It suffices to prove the result for a fixed pair x± ∈ Pa(V ) of index differ-ence one and a fixed parabolic cylinder u ∈ M0(x−, x+;V ). (The assumptionsand conclusions of the theorem are invariant under simultaneous time shift of uand (uε, vε); up to time shift there are only finitely many index one paraboliccylinders with SV ≤ a.) Define

c∗ := SV (x−)− SV (x

+) > 0.

Let (uε, vε) ∈ Mε(x−, x+;V ) with ε ∈ (0, 1]. Denote the time shift of u by

uσ(s, t) := u(s+ σ, t)

for σ ∈ R and define ζ = ζ(σ) = (ξ, η) by

uε = expuσ(ξ), vε = Φ(uσ, ξ) (∂tuσ + η) . (60)

The pair (ξ, η) is well defined whenever σ ‖∂su‖L∞ +‖ξε‖L∞ is smaller than theinjectivity radius ρM of M (i.e. when σ and δε1/2 are sufficiently small). Weassume throughout that δε1/2 ≤ ρM/2 and choose σ0 > 0 so that σ0 ‖∂su‖L∞ <ρM/2.

By Theorem A.1 and Theorem 4.1, there is a constant c0 > 0 such that, forevery ε ∈ (0, 1] and every (uε, vε) ∈ Mε(x−, x+;V ), we have

‖∂su‖∞ + ‖∂tu‖∞ + ‖vε‖∞ ≤ c0. (61)

37

It follows from (60) and (61) that ‖η(σ)‖∞ ≤ c0 for every σ ∈ [−σ0, σ0]. Chooseδ0 > 0 so small that the assertion of the Uniqueness Theorem 3.3 holds withC = c0 and δ = δ0. We shall prove that for every sufficiently small ε > 0 thereis a σ ∈ [−σ0, σ0] such that

ζ(σ) ∈ im(Dεuσ)∗, ‖ξ(σ)‖∞ ≤ δ0ε

1/2, ‖η(σ)‖∞ ≤ c0. (62)

It then follows from Theorem 3.3 that (uε, vε) = T ε(uσ). The proof of (62)will take five steps and uses the following estimate. Choose q > 1 such that1/p+ 1/q = 1. Then, by parabolic exponential decay (see Theorem A.2), thereis a constant c1 > 0 such that, for r = p, q,∞,

‖∂su‖r + ‖∇t∂su‖r + ‖∇s∂su‖r + ‖∇s∇t∂su‖r ≤ c1. (63)

Step 1. For σ ∈ [−σ0, σ0] and ε > 0 sufficiently small define

θε(σ) := −〈Zεσ, ζ〉ε ,

where ζ = ζ(σ) is given by (60) and

Zε :=

(Xε

Y ε

):=

(∂su∇t∂su

)−(ξ∗

η∗

),

ζ∗ :=

(ξ∗

η∗

):= (Dε

u)∗ (Dε

u(Dεu)

∗)−1 Dε

u

(∂su∇t∂su

).

Then θε(σ) = 0 if and only if ζ ∈ im(Dεuσ)∗.

For ε > 0 sufficiently small, the operator Dεu is onto, by Theorem 2.3, and,

by assumption, it has index one (see Remark 2.1). Hence Zε is well definedand belongs to the kernel of Dε

u. It remains to prove that Zε 6= 0 for ε > 0sufficiently small. To see this note that ∂su 6= 0 and so the (0, 2, ε)-norm of thepair (∂su,∇t∂su) is bounded below by a positive constant (the parabolic energyidentity gives c∗ as a lower bound). On the other hand,

ζ∗ = (Dεu)

∗ (Dεu(Dε

u)∗)

−1

(0

∇s∇t∂su

). (64)

Hence, by Theorem 2.3, the (0, 2, ε)-norm of ζ∗ converges to zero as ε tends tozero. It follows that Zε 6= 0 for ε > 0 sufficiently small and this proves Step 1.

Step 2. There are positive constants ε0 and c2 such that

|θε(0)| ≤ c2

(‖ξε‖p + ε2

)

for 0 < ε ≤ ε0 and every (uε = expu(ξε), vε) ∈ Mε(x−, x+;V ) satisfying (59).

We first prove that that there are positive constants ε0 and c3 such that

‖Xε‖q + ‖Y ε‖q ≤ c3 (65)

38

for 0 < ε ≤ ε0. For the summands ∂su of Xε and ∇s∂tu of Y ε this followsfrom (63) with r = q. Moreover, by (64) and Theorem 2.3, we have

‖ξ∗‖q + ε1/2 ‖η∗‖q ≤ c4

(ε ‖(0,∇s∇t∂su)‖0,q,ε + ‖πε(0,∇s∇t∂su)‖q

)

≤ c4ε3/2(ε1/2 + κq) ‖∇s∇t∂su‖q .

The last step uses Lemma C.3 with constant κq > 1. This proves (65). It followsfrom (65) that

|θε(0)| ≤ c3

(‖ξε‖p + ε2 ‖ηε‖p

), (66)

where ηε ∈ Ω0(R× S1, u∗TM) is defined by

vε =: Φ(u, ξε)(∂tuε + ηε).

Define the linear maps Ei(x, ξ) : TxM → Texpx(ξ)M by the formula

d

dτexpx(ξ) =: E1(x, ξ)∂τx+ E2(x, ξ)∇τ ξ (67)

for every smooth path x : R → M and every vector field ξ ∈ Ω0(R, x∗TM)along x. Abbreviate Φ := Φ(u, ξε) and Ei := Ei(u, ξ

ε) for i = 1, 2. Then

ηε = Φ−1vε − ∂tu

= Φ−1(vε − ∂tuε) + Φ−1(E1∂tu+ E2∇tξε)− ∂tu

= ε2Φ−1∇svε +Φ−1E2∇tξ

ε + (Φ−1E1 − 1l)∂tu.

By Corollary 7.2, there is a constant c5 such that ε ‖∇svε‖p ≤ c5. Moreover,

there is a constant c6 > 0 such that∥∥Φ−1E1 − 1l

∥∥p≤ c6 ‖ξε‖p. Hence there is

another constant c7 > 0 such that

‖ηε‖p ≤ c7

(ε+ ‖∇tξ

ε‖p + ‖ξε‖p)≤ c7

(‖ξε‖p + C + 1

).

Combining this with (66) proves Step 2.

Step 3. There is a constant c8 > 0 such that

‖ξ(σ)‖∞ ≤ δε1/2 + c8 |σ| , ‖η(σ)‖∞ ≤ c0,

‖∇sξ‖p ≤ c8, ‖∇σξ + ∂suσ‖p ≤ c8

(|σ|+ δε1/2

), ‖ξ(σ)‖p ≤ δε1/2 + c8|σ|

for 0 < ε ≤ ε0 and |σ| ≤ σ0.

For every σ ∈ R, we have

d (u(s+ σ, t), u(s, t)) ≤ L(γ) ≤ |σ| ‖∂su‖∞ ,

where γ(r) := u(s + rσ, t), 0 ≤ r ≤ 1. Moreover, by (59), d(u(s, t), uε(s, t)) ≤δε1/2. Hence the first estimate of Step 3 follows from the triangle inequality.The second estimate follows from the identity

η(σ) = Φ(uσ, ξ(σ))−1vε − ∂tuσ

39

and (61). To prove the next two estimates we differentiate the identity

expuσ(ξ(σ)) = uε

with respect to σ and s to obtain

E1(uσ, ξ)∂suσ + E2(uσ, ξ)∇σξ = 0, E1(uσ, ξ)∂suσ + E2(uσ, ξ)∇sξ = ∂suε.

By the energy identities the L2 norms of ∂su and ∂suε are uniformly bounded

and hence, so is the L2 norm of ∇sξ. Moreover,

‖∇σξ + ∂suσ‖p =∥∥(E−1

2 E1 − 1l)∂su∥∥p≤ c9 ‖ξ(σ)‖∞ ≤ c10

(|σ|+ δε1/2

).

Hence the Lp norm of ∇σξ is uniformly bounded. Now differentiate the functionσ 7→ ‖ξ(σ)‖p to obtain the inequality ‖ξ(σ)‖p ≤ ‖ξ(0)‖p + c11|σ|. Then the lastinequality in Step 3 follows from (59).

Step 4. Shrinking σ0 and ε0, if necessary, we have

d

dσθε(σ) ≥ c∗

2

for 0 < ε ≤ ε0 and |σ| ≤ σ0.

We will investigate the two terms in the sum

d

dσθε(σ) = − d

dσ〈Xε

σ, ξ(σ)〉 − ε2d

dσ〈Y ε

σ , η(σ)〉 (68)

separately. The key term is 〈Xεσ,∇σξ〉. We have seen that Xε

σ is Lq-close to∂suσ and ∇σξ is Lp-close to −∂suσ. We shall prove that all the other terms aresmall and hence ∂σθ

ε is approximately equal to ‖∂su‖22. More precisely, for thefirst term in (68) we obtain

− d

dσ〈Xε

σ, ξ〉 = −〈Xεσ,∇σξ〉 − 〈∇sX

εσ, ξ〉

= ‖∂su‖22 − 〈Xεσ, ∂suσ +∇σξ〉 − 〈ξ∗, ∂suσ〉

− 〈∇s∂suσ, ξ〉 − 〈ξ∗σ,∇sξ〉

≥ ‖∂su‖22 − c12

(‖∂suσ +∇σξ‖p + ‖ξ∗‖q + ‖ξ‖p

)

≥ ‖∂su‖22 − c13

(|σ|+ δε1/2 + ε3/2

).

Here the second step follows from integration by parts. The third step usesthe inequalities ‖Xε‖q ≤ c (see (65)), ‖∂su‖p + ‖∇s∂su‖q ≤ c (see (63)), and‖∇sξ‖p ≤ c (see Step 3). The last step uses Step 3 and (8).

To estimate the second term in (68) we differentiate the identity

Φ(uσ, ξ(σ))(∂tuσ + η(σ)) = vε

40

with respect to σ to obtain

‖∇s∂tuσ +∇ση‖p ≤ c14

(‖∂su‖p + ‖∇σξ‖p

)≤ c15.

In the first inequality we have used the fact that the L∞ norms of η(σ) and∂tuσ are uniformly bounded. In the second inequality we have used Step 3.Combining this estimate with (63) we find that the Lp norm of ∇ση is uniformlybounded. Differentiating the same identity with respect to s we obtain

‖∇s∂tuσ +∇sη‖p ≤ c16

(‖∇sv

ε‖p + ‖∂su‖p + ‖∇sξ‖p)≤ c17ε

−1.

Here the last inequality follows from Step 3 and Corollary 7.2. Using (63) again,we obtain that the Lp norm of ε∇sη is uniformly bounded. Now

ε2d

dσ〈Y ε

σ , η〉 = ε2〈∇sYεσ , η〉+ ε2〈Y ε

σ ,∇ση〉

= −ε2〈Y εσ ,∇sη〉+ ε2〈Y ε

σ ,∇ση〉≤ c18ε.

In the last estimate we have used (65) and the uniform estimates on the Lp

norms of ∇ση and ε∇sη. Putting things together we obtain

d

dσθε(σ) ≥ ‖∂su‖22 − c19

(|σ|+ ε1/2

).

Since ‖∂su‖22 = c∗, the assertion of Step 4 holds whenever 0 < ε ≤ ε0, |σ| ≤ σ0,

and c19(σ0 + ε1/20 ) ≤ c∗/2.

Step 5. We prove Theorem 8.1.

Suppose the pair (uε, vε) satisfies the requirements of the theorem with ε and δsufficiently small. Then, by Steps 2 and 4, there is a σ ∈ [−σ0, σ0] such that

θε(σ) = 0, |σ| ≤ c20(‖ξε‖p + ε2), c20 :=2c2c∗

.

Let ξ := ξ(σ) and η := η(σ). Then, by Step 3,

‖ξ‖∞ ≤ (δ + c8c20(δ + ε3/2))ε1/2, ‖η‖∞ ≤ c0.

If δ + c8c20(δ + ε3/2) ≤ δ0 then, by Step 1, ζ := (ξ, η) ∈ im (Dεuσ)∗. Hence, by

Theorem 3.3, (uε, vε) = T ε(uσ).

9 Surjectivity

Theorem 9.1. Assume SV is Morse–Smale and fix a constant a ∈ R. Thenthere is a constant ε0 > 0 such that, for every ε ∈ (0, ε0) and every pair x± ∈Pa(V ) of index difference one, the map T ε : M0(x−, x+;V ) → Mε(x−, x+;V ),constructed in Theorem 3.1, is bijective.

41

Lemma 9.2. Assume SV is Morse. Let x± ∈ P(V ) and ui ∈ Mεi(x−, x+;V )where εi is a sequence of positive real numbers converging to zero. Then there isa pair x0, x1 ∈ P(V ), a parabolic cylinder u ∈ M0(x0, x1;V ), and a subsequence,still denoted by (ui, vi), such that the following holds.

(i) ui converges to u strongly in C0 and weakly in W 1,p on every compactsubset of R× S1 and for every p > 1.

(ii) ∂tui and vi converge to ∂tu strongly in Lp and weakly in W 1,2 on everycompact subset of R× S1 and for every p > 1.

(iii) vi − ∂tui converges to zero in the C1 norm on every compact subset ofR× S1.

(iv) For all s ∈ R and T > 0,

limi→∞

AV (ui(s, ·), vi(s, ·)) = SV (u(s, ·)),

limi→∞

∫ T

−T

∫ 1

0

|∂sui|2 dtds =

∫ T

−T

∫ 1

0

|∂su|2 dtds,

limi→∞

ε2i

∫ T

−T

∫ 1

0

|∇svi|2 dtds = 0.

Proof. By Theorems 4.1, 5.1, and 6.1 there is a constant c > 0 such that

‖vi‖∞ + ‖∂tui‖∞ + ‖∂sui‖∞ + ‖∇tvi‖∞ ≤ c, ‖∇svi‖∞ ≤ ε−1i c, (69)

‖∇t∂tui‖∞ ≤ c, ‖∇s∂tui‖p ≤ ε1/p−1/2i c, ‖∇s∂sui‖p ≤ ε

1/p−1i c, (70)

‖∇s∇tvi‖p ≤ ε1/p−1i c, ‖∇s∇svi‖p ≤ ε

−3/2i c (71)

for every i ∈ N and every p ∈ [2,∞]. In (69) the estimate for ∇tvi followsfrom the one for ∂sui and the identity ∇tvi = ∂sui −∇Vt(ui). The estimate for∂tui follows from the ones for vi and ∇svi and the identity ∂tui = vi − ε2i∇svi.In (70) the estimate for ∇s∂sui follows from the one for ∇s∇tvi and the identity∇s∂sui = ∇s∇tvi +∇∂sui∇Vt(ui). The estimate for ∇t∂tui follows from the onesfor ∇tvi and ∇t∇svi and the identity ∇t∂tui = ∇tvi − ε2i∇t∇svi.

By (69), the sequence ui is bounded in C1 and hence in W 1,p([−T, T ]× S1)for every T > 0 and every p > 1. Hence, by the Arzela-Ascoli theorem and theBanach–Alaoglu theorem, a suitable subsequence of ui converges strongly in C0

and weakly in W 1,p on every compact subset of R× S1. By (69) and (70) withp = 2, the sequence ∂tui is bounded in W 1,2([−T, T ] × S1) for every T > 0.Hence, passing to a further subsequence, we may assume that ∂tui convergesstrongly in Lp and weakly in W 1,2 on every compact subset of R × S1 andfor every p > 1. Since vi − ∂tui = ε2i∇svi it follows from (69) and (71) withp = ∞ that vi − ∂tui converges to zero in the C1 norm. Hence vi convergesstrongly in Lp and weakly in W 1,2 on every compact subset of R× S1 and forevery p > 1. This proves that a suitable subsequence of (ui, vi) converges to a

42

function (u, v) : R × S1 → TM as in assertions (i-iii). The limit u is locallyof class W 1,p for every p and ∂tu and v are locally of class W 1,2. By (69), thesequence

vi − ∂tui = ε2i∇svi

converges uniformly to zero, and hence v = ∂tu. Moreover,

∂sui −∇t∂tui −∇Vt(ui) = ε2i∇t∇svi.

By (71), this sequence also converges uniformly to zero. Hence u is a weaksolution of the parabolic partial differential equation

∂su−∇t∂tu−∇Vt(u) = 0.

By the parabolic regularity theorem A.4, u is smooth and so is v = ∂tu.We prove that there is a further subsequence that satisfies (iv). To see

this let us first fix any real number s. Then, by (69) and (70), the sequenceui,s = ui(s, ·) is bounded in C2 and the sequence vi,s = vi(s, ·) is boundedin C1. Hence there exists a further subsequence (depending on s) such thatui,s converges to us in C1 and vi,s converges to vs = ∂tus in C0. Hence, forthis subsequence, AV (us,i, vs,i) converges to SV (us). Now choose any positivesequence Tν → ∞. Then the diagonal subsequence argument shows that thereis a subsequence, still denoted by (ui, vi) such that

limi→∞

AV (uTν ,i, vTν ,i) = SV (uTν ), limi→∞

AV (u−Tν ,i, v−Tν ,i) = SV (u−Tν ),

for every ν. Since ∂sui converges to ∂su weakly in L2 we have

∫ Tν

−Tν

∫ 1

0

|∂su|2 dsdt ≤ limi→∞

∫ Tν

−Tν

∫ 1

0

|∂sui|2 dsdt

≤ limi→∞

∫ Tν

−Tν

∫ 1

0

(|∂sui|2 + ε2i |∇svi|2

)dsdt

= limi→∞

(AV (u−Tν ,i, v−Tν ,i)−AV (uTν ,i, vTν ,i)

)

= SV (u−Tν )− SV (uTν )

=

∫ Tν

−Tν

∫ 1

0

|∂su|2 dsdt

for every ν. This implies (iv) for s = ±Tν and T = Tν . This in turn impliesthat εi∇svi converges to zero in the L2 norm and ∂sui converges to ∂su in theL2 norm on every compact subset of R × S1. Using again the energy-actionidentity on the interval [T, Tν ] for −Tν < T < Tν we deduce that (iv) holdsfor every s and every T . It also follows that the limit u has finite energy.Hence, by Theorem A.2, it belongs to some moduli space M0(x0, x1;V ), wherex0, x1 ∈ P(V ) satisfy

SV (x+) ≤ SV (x1) < SV (x0) ≤ SV (x

−)

This proves the lemma.

43

Lemma 9.3. Assume SV is Morse. Let x± ∈ P(V ) and ui ∈ Mεi(x−, x+;V )where εi is a sequence of positive real numbers converging to zero. Then thereexist periodic orbits x− = x0, x1, . . . , xℓ = x+ ∈ P(V ), parabolic cylinders uk ∈M0(xk−1, xk;V ) for k ∈ 1, . . . , ℓ, a subsequence, still denoted by (ui, vi), andsequences ski ∈ R, k ∈ 1, . . . , ℓ, such that the following holds.

(i) For every k ∈ 1, . . . , ℓ the sequence (s, t) 7→ (ui(ski + s, t), vi(s

ki + s, t))

converges to (uk, ∂tuk) as in Lemma 9.2.

(ii) ski −sk−1i diverges to infinity for k = 2, . . . , ℓ and ∂su

k 6≡ 0 for k = 1, . . . , ℓ.

(iii) For every k ∈ 0, . . . , ℓ and every ρ > 0 there is a constant T > 0 suchthat, for every i and every (s, t) ∈ R× S1,

ski + T ≤ s ≤ sk+1i − T =⇒ d(ui(s, t), x

k(t)) < ρ.

(Here we abbreviate s0i := −∞ and sℓ+1i := ∞.)

Proof. Denote a := SV (x−) and choose ρ > 0 so small that d(x(t), x′(t)) > 2ρ

for every t ∈ R and any two distinct periodic orbits x, x′ ∈ Pa(V ). Choose s1isuch that

sups≤s1i

supt

d(x−(t), ui(s, t)) ≤ ρ, supt

d(x−(t), ui(s1i , t)) = ρ. (72)

Passing to a subsequence we may assume, by Lemma 9.2, that the sequence(ui(s

1i + ·, ·), vi(s1i + ·, ·)) converges in the required sense to a parabolic cylinder

u1 ∈ M0(x0, x1;V ), where x0, x1 ∈ Pa(V ). By (72), we have x0 = x− and x1 6=x0. Hence ∂su

1 6≡ 0 and so SV (x1) < SV (x

0). If x1 = x+ the lemma is proved.If x1 6= x+ choose T > 0 such that d(u1(s, t), x1(t)) < ρ for every t and everys ≥ T . Passing to a subsequence, we may assume that d(ui(s

1i +T, t), x1(t)) < ρ

for every t. Since x1 6= x+ there exists a sequence s2i > s1i + T such that

sups1i+T≤s≤s2i

supt

d(x1(t), ui(s, t)) ≤ ρ, supt

d(x1(t), ui(s2i , t)) = ρ.

The difference s2i − s1i diverges to infinity and, by Lemma 9.2, there is a furthersubsequence such that (ui(s

2i + ·, ·), vi(s2i + ·, ·)) converges to a parabolic cylinder

u2 ∈ M0(x1, x2;V ), where SV (x2) < SV (x

1). Continue by induction. Theinduction can only terminate if xℓ = x+. It must terminate because Pa(V ) is afinite set. This proves the lemma.

Proof of Theorem 9.1. By Theorem 3.1 and Theorem 3.3 the map T ε is welldefined and injective for sufficiently small ε > 0. We will prove surjectivity bycontradiction.

Assume the result is false. Then there exist periodic orbits x± ∈ Pa(V ) ofMorse index difference one and sequences εi > 0 and (ui, vi) ∈ Mεi(x−, x+;V )such that

limi→∞

εi = 0, (ui, vi) /∈ T εi(M0(x−, x+;V )). (73)

44

Applying a time shift, if necessary, we assume without loss of generality that

AV

(ui(0, ·), vi(0, ·)

)=

1

2

(SV (x

−) + SV (x+)). (74)

Fix a constant p > 2. We shall prove in two steps that, after passing to asubsequence if necessary, there is a sequence u0

i ∈ M0(x−, x+;V ) and a constantC > 0 such that

ui = expu0i(ξi),

where the sequence ξi ∈ Ω0(R× S1, (u0i )

∗TM) satisfies

limi→∞

ε−1/2i

(‖ξi‖∞ + ‖ξi‖p

)= 0, ‖∇tξi‖p ≤ C. (75)

Hence it follows from Theorem 8.1 that, for i sufficiently large, there is a realnumber σi such that (ui, vi) = T εi(u0

i (σi+·, ·)). This contradicts (73) and henceproves Theorem 9.1.

Step 1. For every δ > 0 there is a constant T0 > 0 such that

Eεi(ui, vi; (R \ [−T0, T0])× S1) < δ (76)

for every i ∈ N.

Assume, by contradiction, that the statement is false. Then there is a constantδ > 0, a sequence of positive real numbers Ti → ∞, and a subsequence, stilldenoted by (εi, ui, vi), such that, for every i ∈ N,

Eεi(ui, vi; [−Ti, Ti]× S1) ≤ SV (x−)− SV (x

+)− δ. (77)

Choose a further subsequence, still denoted by (ui, vi), that converges as inLemma 9.3 to a finite collection of parabolic cylinders uk ∈ M0(xk−1, xk;V ),k = 1, . . . , ℓ, with x− = x0, x1, . . . , xℓ−1, xℓ = x+ ∈ P(V ). We claim that ℓ ≥ 2.Otherwise, ui(si + ·, ·) converges to u := u1 ∈ M0(x−, x+;V ) as in Lemma 9.2(for some sequence si ∈ R). By (74) and Lemma 9.2 (iv), the sequence si mustbe bounded. By (77), this implies that

E(u; [−T, T ]× S1) =

∫ T

−T

∫ 1

0

|∂su|2 dtds ≤ SV (x−)− SV (x

+)− δ

for every T > 0. This contradicts the fact that u connects x− with x+. Thuswe have proved that ℓ ≥ 2 as claimed. Since SV is Morse–Smale it follows thatthe Morse index difference of x− and x+ is at least two. This contradicts ourassumption and proves Step 1.

Step 2. For i sufficiently large there is a parabolic cylinder u0i ∈ M0(x−, x+;V )

and a vector field ξi ∈ Ω0(R × S1, (u0i )

∗TM) such that ui = expu0i(ξi) and ξi

satisfies (75).

45

Let δ, c and ρ denote the constants in Theorem 7.1 and choose T0 > 0, accordingto Step 1, such that (76) holds with this constant δ. Then, by Corollary 7.2,

|∂sui(s, t)|2 + ε2i |∇svi(s, t)|2 ≤ c3e−ρ|s|Eε(ui, vi; (R \ [−T0, T0])× S1) (78)

for |s| ≥ T0+2 and a suitable constant c3 > 0. By Theorem 4.1 and Theorem 5.1,there is a constant c4 > 0 such that

‖vi‖∞ + ‖∂sui‖∞ + ‖∂tui‖∞ + εi ‖∇svi‖∞ ≤ c4 (79)

for every i. Here we have also used the identity ∂tui = vi − ε2i∇svi. It followsfrom (78) and (79) that there is a constant c5 > 0 such that

|∂sui(s, t)|+ εi |∇svi(s, t)| ≤c5

1 + s2

for every i ∈ N. Moreover, it follows from Theorem 6.1 that

‖∂sui −∇t∂tui −∇Vt(ui)‖p = ε2i ‖∇t∇svi‖p ≤ c6ε1+1/pi .

for a suitable constant c6 > 0. Now let δ0 = δ0(p, c5) and c = c(p, c5) be theconstants in the parabolic implicit function theorem A.5. Then the pair (ui, vi)

satisfies the hypotheses of Theorem A.5, whenever c6ε1+1/pi < δ0. Hence, for i

sufficiently large, there is a parabolic cylinder u0i ∈ M0(x−, x+;V ) and a vector

field ξi ∈ Ω0(R× S1, (u0i )

∗TM) such that

ui = expu0i(ξi),

‖ξi‖Wu0i

≤ c7 ‖∂sui −∇t∂tui −∇Vt(ui)‖p ≤ c6c7ε1+1/pi .

By the Sobolev embedding theorem, we have

‖ξi‖∞ ≤ c8 ‖ξi‖Wu0i

≤ c6c7c8ε1+1/pi

for large i. Moreover, by definition of the Wu0i-norm we have

‖ξi‖p + ‖∇tξi‖p ≤ 2 ‖ξi‖Wu0i

≤ 2c6c7ε1+1/pi .

Hence ξi satisfies (75). This proves Step 2 and the theorem.

Corollary 9.4. Assume SV is Morse–Smale. Then there is a constant ε0 > 0such that the following holds. If 0 < ε ≤ ε0 then the almost complex structureJε ∈ J (T ∗M) of Remark 1.3 is regular in the sense of Floer for the Hamiltonianfunction

HV (t, x, y) :=1

2|y|2 + Vt(x),

and the index one connecting orbits. Moreover,

#M0(x−, x+;V )/R = #Mε(x−, x+;V )/R

for all x± ∈ P(V ) with index difference one.

Proof. By Remark 1.3, the partial differential equation (7) is equivalent to theFloer equation in T ∗M for the pair (HV , Jε). Hence the result follows fromTheorems 3.1 and 9.1.

46

10 Proof of the main result

With Z2 coefficients Theorem 1.1 follows immediately from Corollary 9.4. Toprove the result with general coefficients it remains to examine the orientationsof the moduli spaces. The first step is a result about abstract Fredholm operatorson Hilbert spaces.

Let W ⊂ H be an inclusion of Hilbert spaces that is compact and has a denseimage. Let R 7→ L(W,H) : s 7→ A(s) be a family of bounded linear operatorssatisfying the following conditions.

(A1) The map s 7→ A(s) is continuously differentiable in the norm topology.Moreover, there is a constant c > 0 such that

‖A(s)ξ‖H + ‖A(s)ξ‖H ≤ c ‖ξ‖Wfor every s ∈ R and every ξ ∈ W .

(A2) The operators A(s) are uniformly self-adjoint. This means that, for eachs, the operator A(s), when considered as an unbounded operator on H , isself adjoint, and there is a constant c such that

‖ξ‖W ≤ c (‖A(s)ξ‖H + ‖ξ‖H)

for every s ∈ R and every ξ ∈ W .

(A3) There are invertible operators A± : W → H such that

lims→±∞

∥∥A(s) −A±∥∥L(W,H)

= 0.

(A4) The operator A(s) has only finitely many negative eigenvalues for everys ∈ R.

Denote by S(W,H) the set of invertible self-adjoint operators A : W → H withfinitely many negative eigenvalues. For A ∈ S(W,H) denote by E(A) the directsum of the eigenspaces of A with negative eigenvalues. Given A± ∈ S(W,H)denote by P(A−, A+) the set of functions A : R → L(W,H) that satisfy (A1-4)and by P the union of the spaces P(A−, A+) over all pairs A± ∈ S(W,H). Thisis an open subset of a Banach space.

Denote

W := L2(R,W ) ∩W 1,2(R, H), H := L2(R, H)

and, for every pair A± ∈ S(W,H) and every A ∈ P(A−, A+), consider theoperator DA : W → H defined by

(DAξ)(s) := ξ(s) +A(s)ξ(s)

for ξ ∈ W . This operator is Fredholm and its index is the spectral flow, i.e.

index(DA) = dimE(A−)− dimE(A+)

47

(see Robbin–Salamon [15]). The formal adjoint operator D∗A : W → H is given

by D∗Aη = −η +Aη. Denote by

det(DA) := Λmax (ker DA)⊗ Λmax (ker (DA)∗)

the determinant line of DA and by Or(DA) the set of orientations of det(DA).For A ∈ S(W,H) denote by Or(A) the set of orientations of E(A).

Remark 10.1 (The finite dimensional case). Assume W = H = Rn. LetA± be nonsingular symmetric (n × n)-matrices and A ∈ P(A−, A+). Supposethat A(s) = A± for ±s ≥ T . Define Φ(s, s0) ∈ Rn×n by

∂sΦ(s, s0) +A(s)Φ(s, s0) = 0, Φ(s0, s0) = 1l.

Define

E±(s) :=

ξ ∈ R

n | limr→±∞

Φ(r, s)ξ = 0

.

Then E−(s) = E(A−) for s ≤ −T and E+(A) = E(A+)⊥ for s ≥ T . Moreover,

kerDA∼= E−(s) ∩ E+(s), (imDA)

⊥ ∼= (E−(s) + E+(s))⊥.

Hence there is a natural map

τA : Or(A−)×Or(A+) → Or(DA)

defined as follows. Given orientations of E(A−) ∼= E−(s) and E(A+) ∼= E+(s)⊥,pick any basis u1, . . . , uℓ of E−(s) ∩ E+(s) ∼= ker DA. Extend it to a positivebasis of E−(s) by picking a suitable basis v1, . . . , vm of E−(s) ∩ E+(s)⊥. Nowextend the vectors vj to a positive basis of E+(s)⊥ by picking a suitable ba-sis w1, . . . , wn of (E−(s) + E+(s))⊥ ∼= (imDA)

⊥. Then the bases u1, . . . , uℓ

of kerDA and w1, . . . , wn of (imDA)⊥ determine the induced orientation of

det(DA). Note that this is well defined (a sign change in the ui leads to a signchange in the wk).

Remark 10.2 (Catenation). Let A0, A1, A2 ∈ S(W,H) and suppose thatA01 ∈ P(A0, A1) and A12 ∈ P(A1, A2) satisfy

A01(s) =

A0 if s ≤ −T,

A1 if s ≥ T,A12(s) =

A1 if s ≤ −T,

A2 if s ≥ T.(80)

For R > T define AR02 ∈ P(A0, A2) by

AR02(s) =

A01(s+R) if s ≤ 0,

A12(s−R) if s ≥ 0.(81)

If DA01and DA12

are onto then, for R sufficiently large, the operator DAR02

isonto and there is a natural isomorphism

SR : ker DA01⊕ ker DA12

→ ker DAR02

48

The isomorphism SR is defined by composing a pre-gluing operator with theorthogonal projection onto the kernel. That this gives an isomorphism followsfrom exponential decay estimates for the elements in the kernel and a uniformestimate for suitable right inverses of the operators DAR

02(see for example [16]).

Theorem 10.3. There is a family of maps

τA : Or(A−)×Or(A+) → Or(DA),

one for each pair of Hilbert spaces W ⊂ H with a compact dense inclusion, eachpair A± ∈ S(W,H), and each A ∈ P(A−, A+), satisfying the following axioms.

(Equivariant) τA is equivariant with respect to the Z2-action on each factor.

(Homotopy) The map (A, o−, o+)7→(A, τA(o−, o+)) from the topological space

(A, o−, o+) |A ∈ P , o± ∈ Or(A±) to (A, o) |A ∈ P , o ∈ Or(DA) iscontinuous.

(Naturality) Let Φ(s) : (W,H) → (W ′, H ′) be a family of (pairs of) Hilbertspace isomorphisms that is continuously differentiable in the operator normon H and continuous in the operator norm on W . Suppose that thereexist Hilbert space isomorphisms Φ± : (W,H) → (W ′, H ′) such that Φ(s)converges to Φ± in the operator norm on both spaces and Φ(s) convergesto zero in L(H) as s → ±∞. Then

τΦ∗A(Φ−∗ o

−,Φ+∗ o

+) = Φ∗τA(o−, o+)

for all A± ∈ S(W,H), A ∈ P(A−, A+), and o± ∈ Or(A±).

(Direct Sum) If A±j ∈ S(Wj , Hj) and Aj ∈ P(A−

j , A+j ) for j = 0, 1 then

τA0⊕A1(o−0 ⊗ o−1 , o

+0 ⊗ o+1 ) = τA0

(o−0 , o+0 )⊗ τA1

(o−1 , o+1 ).

for all o±j ∈ Or(A±j ).

(Catenation) Let A0, A1, A2 ∈ S(W,H), suppose that A01 ∈ P(A0, A1) andA12 ∈ P(A1, A2) satisfy (80) and, for R > T , define AR

02 by (81). AssumeDA01

and DA12are onto. Then DAR

02is onto for large R and

τA02(o0, o2) = σR (τA01

(o0, o1), τA12(o1, o2)) .

for o0 ∈ Or(A0), o1 ∈ Or(A1), and o2 ∈ Or(A2). Here the map

σR : det(DA01)× det(DA12

) → det(DAR02)

is induced by the isomorphism SR of Remark 10.2.

(Constant) If A(s) ≡ A+ = A− and o+ = o− ∈ Or(A±) then τA(o−, o+) is

the standard orientation of det(DA) ∼= R.

49

(Normalization) If W = H = Rn then τA is the map defined in Remark 10.1.

The maps τA are uniquely determined by the (Homotopy), (Direct Sum), (Con-stant), and (Normalization) axioms.

Theorem 10.3 is standard with the techniques of [7] (although the assump-tions are not quite the same as in the work of Floer and Hofer).

Proof of Theorem 1.1. Assume SV is Morse–Smale. For x ∈ P(V ) denote byWu(x) the unstable manifold of x with respect to the negative gradient flow ofSV . Thus Wu(x) is the space of all smooth loops y : S1 → M such that thereexists a solution u : (−∞, 0]× S1 → M of the nonlinear heat equation (3) thatconverges to x as s → −∞ and satisfies u(0, t) = y(t). Then Wu(x) is a finitedimensional manifold (see for example [2]). It is diffeomorphic to R

k wherek = indV (x) is the Morse index of x as a critical point of SV . Fix an orientationof Wu(x) for every periodic orbit x ∈ P(V ). These orientations determine asystem of coherent orientations for the heat flow as follows.

Fix a pair x± ∈ P(V ) of periodic orbits that represent the same componentof LM . Denote by P0(x−, x+) the set of smooth maps u : R × S1 → M suchthat u(s, ·) converges to x± in the C2 norm and ∂su(s, ·) converges to zero inthe C1 norm as s tends to ±∞. Then, in a suitable trivialization of the tangentbundle u∗TM , the linearized operator D0

u has the form of an operator DA asin Theorem 10.3 where the spaces E(A±) correspond to the tangent spacesTx±Wu(x±) of the unstable manifolds. Hence, by Theorem 10.3, the givenorientations of the unstable manifolds determine orientations

ν0(u) ∈ Or(det(D0u))

of the determinant lines for all u ∈ P0(x−, x+) and all x± ∈ P(V ). By the(Naturality) axiom, these orientations are independent of the choice of the triv-ializations used to define them. By the (Catenation) axiom, they form a systemof coherent orientations in the sense of Floer–Hofer [7].

Next we show how the coherent orientations for the heat flow induce a systemof coherent orientations

νε(u, v) ∈ Or(det(Dεu,v))

for the Floer equations of the pair (HV , Jε). Let us denote by P(x−, x+) theset of smooth maps (u, v) : R × S1 → TM such that (u(s, ·), v(s, ·)) convergesto (x±, x±) in the C1 norm and (∂su,∇sv) converges to zero, uniformly in t,as s tends to ±∞. By the obvious homotopy arguments it suffices to assumeu ∈ P0(x−, x+) and v = ∂tu. We abbreviate Dε

u := Dεu,∂tu

. It follows from thedefinition of the operators in (10) that

D0uξ = 0 =⇒ Dε

u

(ξ∇tξ

)=

(0

∇s∇tξ +R(ξ, ∂su)∂tu

).

50

Hence Dεu(ξ,∇tξ) is small in the (0, 2, ε)-norm. If the operator D0

u is onto thenthe estimate of Theorem 2.3 shows that the map

ker D0u → ker Dε

u : ξ 7→(

ξ∇tξ

)−Dε

u∗ (Dε

uDεu∗)

−1 Dεu

(ξ∇tξ

)

is an isomorphism between the kernels and we define νε(u, ∂tu) to be the imageof ν0(u) under the induced isomorphism of the top exterior powers. If D0

u isnot onto we obtain a similar isomorphism between the determinant lines of D0

u

and Dεu by augmenting the operators first to make them surjective. It follows

again from the (Catenation) axiom that the νε(u, v) form a system of coherentorientations for the Floer equations.

Now assume that x± ∈ P(V ) have Morse index difference one. Consider themap

T ε : M0(x−, x+;V ) → Mε(x−, x+;V )

defined in Theorem 3.1 and recall that, by Theorem 9.1, it is bijective. Itfollows from the proof of Theorem 8.1 that the map T ε satisfies the following.Let u ∈ M0(x−, x+;V ) and (uε, vε) := T ε(u) ∈ Mε(x−, x+;V ). Then thevector ∂su ∈ ker D0

u is positively oriented with respect to ν0(u) if and onlyif the vector (∂su

ε,∇svε) ∈ kerDε

uε,vε is positively oriented with respect toνε(uε, vε). Hence the bijection T ε preserves the signs for the definitions of thetwo boundary operators. This shows that the Morse complex of the heat flowhas the same boundary operator as the Floer complex for ε sufficiently small.Hence the resulting homologies are naturally isomorphic, i.e. for every a ∈ R

there is a constant ε0 > 0 such that

HFa∗(T

∗M,HV , Jε) ∼= HMa∗(LM,SV )

for 0 < ε ≤ ε0. In fact, we have established this isomorphism on the chain leveland with integer coefficients. This proves Theorem 1.1.

A The heat flow

In this appendix we summarize results from [22] that are used in this paper. Weassume throughout this appendix that M is a compact Riemannian manifold.Let S1 ×M → R : (t, x) 7→ Vt(x) be a smooth potential and consider the heatequation

∂su−∇t∂tu−∇Vt(u) = 0 (82)

for smooth functions R× S1 → M : (s, t) 7→ u(s, t).

Theorem A.1 (Apriori estimates). Fix a potential V : S1 × M → R anda constant c0 > 0. Then there is a constant C = C(c0, V ) > 0 such that thefollowing holds. If u : R×S1 → M is a solution of (82) such that SV (u(s, ·)) ≤ c0for every s ∈ R then

‖∂su‖∞ + ‖∂tu‖∞ + ‖∇t∂tu‖∞ ≤ C.

51

Theorem A.2 (Exponential decay). Suppose SV is Morse.

(F) Let u : [0,∞) × S1 → M be a solution of (82). Then there are positiveconstants ρ and c1, c2, c3, . . . such that

‖∂su‖Ck([T,∞)×S1) ≤ cke−ρT

for every T ≥ 1. Moreover, there is a periodic orbit x ∈ P(V ) such thatu(s, t) converges to x(t) as s → ∞.

(B) Let u : (−∞, 0]× S1 → M be a solution of (82) with finite energy

∫ 0

−∞

∫ 1

0

|∂su(s, t)|2 dtds < ∞.

Then there are positive constants ρ and c1, c2, c3, . . . such that

‖∂su‖Ck((−∞,−T ]×S1) ≤ cke−ρT

for every T ≥ 1. Moreover, there is a periodic orbit x ∈ P(V ) such thatu(s, t) converges to x(t) as s → −∞.

Theorem A.3 (Fredholm). Let x± ∈ P(V ) be nondegenerate. Moreover, letu : R × S1 → M be a smooth map such that u(s, ·) converges to x± in the C2

norm and ∂su converges uniformly to zero as s → ±∞. Then, for every p > 1,the operator D0

u : Wpu → Lp

u is Fredholm and its Fredholm index is given by

indexD0u = indV x(x

−)− indV (x+).

Here indV (x±) denotes the Morse index of x±. Furthermore,

ker D0u = coker (D0

u)∗, ker (D0

u)∗ = cokerD0

u.

Theorem A.4 (Regularity). Fix a constant p > 2. Let u : R× S1 → M be acontinuous function which is locally of class W 1,p. Assume further that ∂tu islocally of class W 1,2 and that u is a weak solution of (82). Then u is smooth.

Theorem A.5 (Implicit function theorem). Let SV be Morse-Smale andx± ∈ P(V ) of index difference one. Then, for all c0 > 0 and p > 2, there existpositive constants δ0 and c such that the following holds. If u : R × S1 → Mis a smooth map such that lims→±∞ u(s, ·) = x±(·) exists, uniformly in t, andsuch that

|∂su(s, t)| ≤c0

1 + s2, ∀(s, t), ‖∂tu‖∞ ≤ c0, ‖F0(u)‖p ≤ δ0.

Then there exist elements u0 ∈ M0(x−, x+;V ) and ξ ∈ imD0u0

∗∩Wu0satisfying

u = expu0ξ, ‖ξ‖Wu0

≤ c‖F0(u)‖p.

Here F0(u) := ∂su−∇t∂tu−∇Vt(u).

52

Let us defineVreg ⊂ C∞(S1 ×M)

as the subset of those potentials V for which SV is Morse-Smale; i.e. for whichall x ∈ P(V ) are nondegenerate and for which the operator D0

u is surjective forevery pair x± ∈ P(V ) and every connecting orbit u ∈ M0(x−, x+;V ). A subsetof a complete metric space is called residual if it contains a countable intersectionof open and dense sets. By Baire’s category theorem a residual subset is dense.We emphasize that C∞(S1 ×M) has the topology of a complete metric space.

Theorem A.6 (Transversality). The set Vreg is residual in C∞(S1 ×M).

Theorem A.7. For every regular value a of SV there is a natural isomorphism

HMa∗ (LM,SV ) ∼= H∗(LaM,Z).

If M is not simply connected then there is a separate isomorphism for eachcomponent of the loop space.

B Two fundamental Lp estimates

Theorem B.1. For every p > 1 there is a constant c = c(p) > 0 such that

‖∂su‖Lp + ‖∂sv‖Lp ≤ c (‖∂su− ∂tv‖Lp + ‖∂sv + ∂tu− v‖Lp) (83)

for all u, v ∈ C∞0 (R2).

Theorem B.2. For every p > 1 there is a constant c = c(p) > 0 such that

‖∂su‖Lp + ‖∂t∂tu‖Lp ≤ c ‖∂su− ∂t∂tu‖Lp (84)

for every u ∈ C∞0 (R2).

If we assume v = ∂tu then (83) follows from (84) (but not conversely). Onthe other hand if the term ∂sv + ∂tu− v on the right is replaced by ∂sv + ∂tu,then (83) becomes the Calderon-Zygmund inequality. However, it seems that theestimate (83) in its full strength cannot be deduced directly from the Calderon–Zygmund inequality and the parabolic estimate (84). Theorems B.1 and B.2will be proved below.

Corollary B.3. Let p > 1 and denote by c = c(p) the constant of Theorem B.1.Then

‖∂su‖Lp + ε ‖∂sv‖Lp ≤ c(‖∂su− ∂tv‖Lp + ε

∥∥∂sv + ε−2(∂tu− v)∥∥Lp

)(85)

for every ε > 0 and every pair u, v ∈ C∞0 (R2).

53

Proof. Denote

f := ∂su− ∂tv, g := ∂sv + ε−2(∂tu− v).

Now consider the rescaled functions

u(s, t) := u(ε2s, εt), v(s, t) := εv(ε2s, εt)

andf(s, t) := ε2f(ε2s, εt), g(s, t) := ε3g(ε2s, εt).

Then∂su− ∂tv = f , ∂sv + ∂tu− v = g.

Hence, by Theorem B.1,∥∥∂su

∥∥Lp +

∥∥∂sv∥∥Lp ≤ c

(∥∥f∥∥Lp +

∥∥g∥∥Lp

).

Now the result follows from the fact that∥∥∂su

∥∥Lp = ε2−3/p

∥∥∂su∥∥Lp ,

∥∥f∥∥Lp = ε2−3/p

∥∥f∥∥Lp ,

and similarly for the other terms.

We give a proof of (83) and (84) that is based on the Marcinkiewicz–Mihlinmultiplier method. To formulate the result, we consider the Fourier transform

F : L2(R2,C) → L2(R2,C),

given by

(Ff)(σ, τ) :=1

2π

∫ ∞

−∞

∫ ∞

−∞

e−i(σs+τt)f(s, t) dsdt

for f ∈ L2(R2,C) ∩ L1(R2,C). Given a bounded measurable complex valuedfunction m : R2 → C define the bounded linear operator

Tm : L2(R2,C) → L2(R2,C)

byTmf := F−1(mFf).

The following theorem is proved in [12].

Theorem B.4 (Marcinkiewicz–Mihlin). For every c > 0 and every p > 1there is a constant cp = cp(c) > 0 such that the following holds. If m : R2 → C

is a measurable function such that the restriction of m to each of the four openquadrants in R

2 is twice continuously differentiable and

|m(σ, τ)| + |σ∂σm(σ, τ)| + |τ∂τm(σ, τ)| + |στ∂σ∂τm(σ, τ)| ≤ c (86)

for σ, τ ∈ R \ 0 then

f ∈ Lp(R2,C) ∩ L2(R2,C) =⇒ Tmf ∈ Lp(R2,C)

and‖Tmf‖Lp ≤ cp ‖f‖Lp

for every f ∈ Lp(R2,C) ∩ L2(R2,C).

54

Remark B.5. The theorem of Marcinkiewicz–Mihlin in its original form isslightly stronger than Theorem B.4, namely condition (86) is replaced by theweaker conditions

supσ,τ

|m(σ, τ)| ≤ c, (87)

supσ 6=0

∫ 2ℓ+1

2ℓ|∂τm(σ,±τ)| dτ ≤ c, sup

τ 6=0

∫ 2k+1

2k|∂σm(±σ, τ)| dσ ≤ c (88)

and ∫ 2k+1

2k

∫ 2ℓ+1

2ℓ|∂σ∂τm(±σ,±τ)| dτ ≤ c (89)

for all integers k and ℓ (and all choices of signs). In this form the result is provedin Stein [18, Theorem 6’]. It is easy to see that (86) implies (88) with c replacedby c log 2 and (89) with c replaced by c(log 2)2.

Proof of Theorem B.2. Let u ∈ C∞0 (R2,C) and define f ∈ C∞

0 (R2) by

f := ∂su− ∂t∂tu.

Denote the Fourier transforms of f and u by

f := Ff, u := Fu.

Thenf = iσu+ τ2u

and hence

∂su = iσu =iσ

τ2 + iσf .

Denote the multiplier in this equation by

m(σ, τ) :=iσ

τ2 + iσ.

The formulae

∂σm =iτ2

(τ2 + iσ)2 , ∂τm =

−2iστ

(τ2 + iσ)2 , ∂σ∂τm =

−2iτ(τ2 − iσ)

(τ2 + iσ)3

show that the functions m, σ∂σm, τ∂τm, and στ∂σ∂τm are bounded. Hencethe result follows from Theorem B.4.

Proof of Theorem B.1. Let u, v ∈ C∞0 (R2,C) and define f, g ∈ C∞

0 (R2) by

f := ∂su− ∂tv, g := ∂sv + ∂tu− v.

Thenf = iσu− iτ v, g = iσv + iτ u− v.

55

Solving this equation for u and v we find

u =1− iσ

σ2 + τ2 + iσf − iτ

σ2 + τ2 + iσg,

v =iτ

σ2 + τ2 + iσf − iσ

σ2 + τ2 + iσg,

and hence

∂su =σ2 + iσ

σ2 + τ2 + iσf +

στ

σ2 + τ2 + iσg,

∂sv =−στ

σ2 + τ2 + iσf +

σ2

σ2 + τ2 + iσg.

The four multipliers in the last two equations satisfy (86). Hence the resultfollows from Theorem B.4.

C The estimate for the inverse

We begin by proving a weaker version of the estimate in Theorem 2.2.

Proposition C.1. Let u ∈ C∞(R × S1,M) and v ∈ Ω0(R × S1, u∗TM) suchthat ‖∂su‖∞, ‖∂tu‖∞ and ‖v‖∞ are finite and lims→±∞ u(s, t) exists, uniformlyin t. Then, for every p > 1, there is a constant c > 0 such that

ε−1‖∇tξ − η‖p + ‖∇tη‖p + ‖∇sξ‖p + ε‖∇sη‖p≤ c

(‖Dε

u,vζ‖0,p,ε + ε−1‖ζ‖0,p,ε) (90)

for every ε ∈ (0, 1] and every pair of compactly supported vector fields ζ =(ξ, η) ∈ Ω0(R × S1, u∗TM ⊕ u∗TM). The formal adjoint operator (Dε

u,v)∗ sat-

isfies the same estimate.

Proof. Choose a finite open cover Uαα of the cylinder R×S1 with the followingproperties.

(i) For each α the set Uα ⊂ R× S1 is contractible.

(ii) For each α the closure of the image of Uα under u is contained in acoordinate chart on M .

(iii) There is a constant T > 0 and an open cover Iαα of S1 such thatUα ∩ [T,∞) × S1 = [T,∞) × Iα for every α. Similarly for the interval(−∞,−T ].

We prove (90) for Dεu,v. The estimate for (Dε

u,v)∗ is analoguous. Assume first

that ξ and η are compactly supported in Uα for some α and denote by ξα, ηα :Uα → Rn the vector fields in local coordinates. By Corollary B.3, there is aconstant cα, depending only on p and the metric, such that

‖∂sξα‖p + ε ‖∂sηα‖p ≤ cα

(‖∂sξα − ∂tηα‖p + ε

∥∥∂sηα + ε−2(∂tξα − ηα)∥∥p

).

56

Here we denote by ‖·‖p the Lp norm with respect to the Riemannian metric inthe coordinate charts on M . Replacing the partial derivatives ∂s and ∂t by thecovariant derivatives ∇s and ∇t we obtain

‖∇sξ‖p + ε ‖∇sη‖p ≤ c(‖∇sξ −∇tη‖p + ε

∥∥∇sη + ε−2(∇tξ − η)∥∥p

+ ε−1 ‖ξ‖p + ‖η‖p).

(91)

for every ξ with support in one of the sets Uα. Here we have used the L∞ boundson ∂su and ∂tu. Observe that the constant c depends on the Christoffel symbolsdetermined by our coordinate chart onM . Now let βαα be a partition of unitysubordinate to the cover Uαα such that ‖∂sβα‖∞+‖∂tβα‖∞ < ∞ for every α.(Note that βα need not have compact support when Uα is unbounded.) Givenany two compactly supported vector fields ξ, η ∈ Ω0(R×S1, u∗TM) apply (91) tothe (compactly supported) pair (βαξ, βαη) and take the sum to deduce that (91)continues to hold for the pair (ξ, η) with an appropriate larger constant c. Usingthe L∞ bounds on ∂su, ∂tu, and v (as well as on the curvature and the secondderivatives of Vt) we obtain

‖∇sξ‖p + ε ‖∇sη‖p ≤ c′(‖∇sξ −∇tη −R(ξ, ∂tu)v −∇ξ∇Vt(u)‖p+ ε

∥∥∇sη +R(ξ, ∂su)v + ε−2(∇tξ − η)∥∥p

+ ε−1 ‖ξ‖p + ‖η‖p).

This implies (90).

Under the assumptions of Proposition C.1 it follows immediately that

‖ζ‖1,p,ε ≤ c(ε2‖Dε

u,vζ‖0,p,ε + ‖ζ‖0,p,ε)

(92)

and similarly for (Dεu,v)

∗. Moreover, note that the difference between Proposi-tion C.1 and Theorem 2.2 lies in the ε-factors in front of ‖ξ‖p and ‖η‖p on theright hand sides of the estimates. To prove Theorem 2.2 we must improve thesethese factors by ε for ξ and by ε2 for η. This requires the following parabolicestimate. Let 1/p+ 1/q = 1. The formal adjoint operator

(D0u)

∗ : Wqu → Lq

u

of D0u : Wp

u → Lpu is given by

(D0u)

∗ξ = −∇sξ −∇t∇tξ −R(ξ, ∂tu)∂tu−∇ξ∇V (t, u). (93)

Proposition C.2. Let u ∈ C∞(R × S1,M) such that ‖∂su‖∞, ‖∂tu‖∞ and‖∇t∂tu‖∞ are finite and lims→±∞ u(s, t) exists, uniformly in t. Then, for everyp > 1, there is a constant c > 0 such that

‖∇sξ‖p + ‖∇t∇tξ‖p ≤ c(‖D0

uξ‖p + ‖ξ‖p)

(94)

for every compactly supported vector field ξ ∈ Ω0(R × S1, u∗TM). The formaladjoint operator (D0

u)∗ satisfies the same estimate.

57

Lemma C.3. Let x : S1 → M be a smooth map, p > 1 and

κp :=

p if p ≥ 2,

p/(p− 1) if p ≤ 2.(95)

Then, for every ε > 0 and every ξ ∈ Ω0(S1, x∗TM), we have

‖(1l− ε∇t∇t)−1ξ‖p ≤ ‖ξ‖p,√

ε‖(1l− ε∇t∇t)−1∇tξ‖p ≤ κp‖ξ‖p,

ε‖(1l− ε∇t∇t)−1∇t∇tξ‖p ≤ 2‖ξ‖p.

These estimates continue to hold for u ∈ C∞(R × S1,M) and compactly sup-ported vector fields ξ ∈ Ω0(R× S1, u∗TM).

Proof. First consider the case p ≥ 2: Let ε > 0 and ξ ∈ Ω0(S1, x∗TM). Define

η := (1l− ε∇t∇t)−1ξ.

(The operator (1l − ε∇t∇t) : W 2,p(S1, x∗TM) → Lp(S1, x∗TM) is bijective.)Then

d2

dt2|η|p =

d

dt

(p |η|p−2 〈∇tη, η〉

)

= p(p− 2)|η|p−4〈∇tη, η〉2 + p|η|p−2(〈∇t∇tη, η〉+ |∇tη|2

)

≥ pε−1|η|p − pε−1|η|p−2〈ξ, η〉≥ pε−1|η|p − pε−1|η|p−1|ξ|≥ ε−1|η|p − ε−1|ξ|p.

The third step uses the identity ∇t∇tη = ε−1η−ε−1ξ. The last step uses Young’sinequality

ab ≤ ar

r+

bs

s,

1

r+

1

s= 1, (96)

with r = p, a = |ξ| and s = p/(p− 1), b = |η|p−1. Moreover,

d

dt

(|∇tη|p−2 〈∇tη, η〉

)

= |∇tη|p + |∇tη|p−2〈∇t∇tη, η〉+ (p− 2)|∇tη|p−4〈∇tη, η〉〈∇t∇tη,∇tη〉= |∇tη|p + ε−1|∇tη|p−2|η|2 − ε−1|∇tη|p−2〈ξ, η〉− ε−1(p− 2)|∇tη|p−4〈∇tη, η〉〈ξ,∇tη〉+ ε−1(p− 2)|∇tη|p−4〈∇tη, η〉2

≥ |∇tη|p + 12ε

−1|∇tη|p−2|η|2 − p−12 ε−1|∇tη|p−2|ξ|2 + p−2

2 ε−1|∇tη|p−4〈∇tη, η〉2

≥ |∇tη|p − p−12 ε−1|∇tη|p−2|ξ|2

≥ 2p |∇tη|p − 2

p

(p−12

)p/2ε−p/2|ξ|p.

58

The third step uses (96) with r = s = 2. The last step uses (96) with r = p/2,a = p−1

2 ε−1|ξ|2 and s = p/(p− 2), b = |∇tη|p−2. Now the first two estimates ofthe lemma follow by integration over S1, respectively R×S1. The last estimateis an easy consequence of the first:

ε‖∇t∇tη‖p = ‖η − ξ‖p ≤ ‖η‖p + ‖ξ‖p ≤ 2‖ξ‖p.

This proves the lemma for p ≥ 2. Now assume 1 < p < 2 and let q := p/(p− 1).Then q > 2 and hence

√ε∥∥(1l− ε∇t∇t)

−1∇tξ∥∥p=

√ε sup06=η∈Lq

⟨(1l− ε∇t∇t)

−1∇tξ, η⟩

‖η‖q

≤√ε sup06=η∈Lq

∥∥ξ∥∥p

∥∥(1l− ε∇t∇t)−1∇tη

∥∥q

‖η‖q≤ q ‖ξ‖p .

This prove the second estimate for p < 2. The other estimates follow similarly.This proves the lemma.

Lemma C.4. Let x ∈ C∞(S1,M) and p > 1. Then

‖∇tξ‖p ≤ κp

(δ−1‖ξ‖p + δ‖∇t∇tξ‖p

)

for δ > 0 and ξ ∈ Ω0(S1, x∗TM), where κp is defined by (95). This estimatecontinues to hold for u ∈ C∞(R× S1,M) and compactly supported vector fieldsξ ∈ Ω0(R× S1, u∗TM).

Proof. Let 1/p+ 1/q = 1. Since the operator

W 2,q(S1, x∗TM) → Lq(S1, x∗TM) : η 7→ δ−1η + δ∇t∇tη

is bijective, we have

‖∇tξ‖p = supη∈W 2,q

〈∇tξ, δ−1η − δ∇t∇tη〉

‖δ−1η − δ∇t∇tη‖q

= supη∈W 2,q

−〈ξ, δ−1∇tη〉+ 〈∇t∇tξ, δ∇tη〉‖δ−1η − δ∇t∇tη‖q

≤(δ−1 ‖ξ‖p + δ ‖∇t∇tξ‖p

)sup

η∈W 2,q

‖∇tη‖q‖δ−1η − δ∇t∇tη‖q

≤ κp

(δ−1 ‖ξ‖p + δ ‖∇t∇tξ‖p

).

To prove the last step, denote

ζ := η − δ2∇t∇tη.

Then∇tη =

(1l− δ2∇t∇t

)−1 ∇tζ

59

and hence, by Lemma C.3 with ε = δ2, we have

‖∇tη‖q ≤ κqδ−1 ‖ζ‖q = κp

∥∥δ−1η − δ∇t∇tη∥∥q.

We have used the fact that κp = κq. This proves the lemma.

Proof of Proposition C.2. The proof follows the same pattern as that of Propo-sition C.1. Let Uαα be as above. If ξ is (compactly) supported in Uα then,by Theorem B.2,

‖∂sξα‖p + ‖∂t∂tξα‖p ≤ cα ‖∂sξα − ∂t∂tξα‖p

Replacing ∂s and ∂t by ∇s and ∇t, and using the L∞ bounds on ∂su, ∂tu, and∇t∂tu, we find

‖∇sξ‖p + ‖∇t∇tξ‖p ≤ c(‖∇sξ −∇t∇tξ‖p + ‖ξ‖p + ‖∇tξ‖p

).

Using a partition of unity βαα, subordinate to the cover Uαα, such that

‖∂sβα‖∞ + ‖∂tβα‖∞ + ‖∂t∂tβα‖∞ < ∞,

we deduce that the last estimate continues to hold for every compactly supportedvector field ξ ∈ Ω0(R × S1, u∗TM). Now apply Lemma C.4 with δcp < 1/2 toobtain

‖∇sξ‖p + ‖∇t∇tξ‖p ≤ c′(‖∇sξ −∇t∇tξ‖p + ‖ξ‖p

).

Hence

‖∇sξ‖p + ‖∇t∇tξ‖p ≤ c′′(‖∇sξ −∇t∇tξ −R(ξ, ∂tu)∂tu−∇ξ∇Vt(u)‖p + ‖ξ‖p

)

as required.

Proof of Theorem 2.2. Fix a constant p > 1 and define

f(ξ, η) := ∇sξ −∇tη, g(ξ, η) := ∇sη + ε−2(∇tξ − η),

for compactly supported vector fields ζ = (ξ, η) ∈ Ω0(R× S1, u∗TM ⊕ u∗TM).It suffices to show that

ε−1 ‖∇tξ − η‖p + ‖∇tη‖p + ‖∇sξ‖p + ε ‖∇sη‖p≤ c

(‖f‖p + ε ‖g‖p + ‖ξ‖p + ε2 ‖η‖p

) (97)

for some constant c > 0 independent of ε and (ξ, η). The general case (for Dεu,v)

then follows easily:

ε−1 ‖∇tξ − η‖p + ‖∇tη‖p + ‖∇sξ‖p + ε ‖∇sη‖p≤ c′

(‖f −R(ξ, ∂tu)v −∇ξ∇Vt(u)‖p + ε ‖g +R(ξ, ∂su)v‖p + ‖ξ‖p + ε2 ‖η‖p

).

60

To prove the estimate for the formal adjoint operator (Dεu,v)

∗ apply (97) to thevector fields ξ(−s, t) and η(−s, t) and then proceed as above.

To prove (97) we split ζ into two components. Let

πε(ξ, η) := (1l− ε∇t∇t)−1(ξ − ε2∇tη), ι(ξ) := (ξ,∇tξ),

and define

ζ0 :=

(ξ0η0

):= ιπεζ =

((1l− ε∇t∇t)

−1(ξ − ε2∇tη)∇t(1l− ε∇t∇t)

−1(ξ − ε2∇tη)

),

ζ1 :=

(ξ1η1

):= ζ − ζ0 =

((1l− ε∇t∇t)

−1(ε2∇tη − ε∇t∇tξ)(1l− ε∇t∇t)

−1(η −∇tξ + (ε2 − ε)∇t∇tη)

).

Note that η0 = ∇tξ0 and

ξ1 − ε∇tη1 = (ε2 − ε)∇tη. (98)

Since f and g are linear, we obtain the splitting f = f0 + f1 and g = g0 + g1,where fi := f(ξi, ηi) and gi := g(ξi, ηi) for i = 0, 1. Thus

f0 = ∇sξ0 −∇t∇tξ0, g0 = ∇s∇tξ0.

Now apply the parabolic estimate of Proposition C.2, with a constant c0 > 0,to ξ0 and the elliptic estimate of Proposition C.1, with a constant c1 > 0, to(ξ1, η1). This gives

ε−1 ‖∇tξ − η‖p + ‖∇tη‖p + ‖∇sξ‖p + ε ‖∇sη‖p≤ ‖∇t∇tξ0‖p + ‖∇sξ0‖p + ε ‖∇s∇tξ0‖p+ ε−1 ‖∇tξ1 − η1‖p + ‖∇tη1‖p + ‖∇sξ1‖p + ε ‖∇sη1‖p

≤ c0(‖f0‖p + ‖ξ0‖p

)+ ε ‖g0‖p

+ c1(‖f1‖p + ε ‖g1‖p + ε−1 ‖ξ1‖p + ‖η1‖p

)

≤ c1(‖f‖p + ε ‖g‖p + ε−1 ‖ξ1‖p + ‖η1‖p

)

+ (c0 + c1) ‖f0‖p + (1 + c1)ε ‖g0‖p + c0 ‖ξ0‖p .

(99)

We examine the last five terms on the right individually. For this we shall needthe commutator identities

[∇s,∇t] = R(∂su, ∂tu), (100)

[∇s,∇t∇t] = 2∇t[∇s,∇t] + (∇tR)(∂tu, ∂su)

+ R(∇t∂tu, ∂su) +R(∂tu,∇t∂su),(101)

[∇s, (1l− ε∇t∇t)−1] = (1l− ε∇t∇t)

−1[1l− ε∇t∇t,∇s](1l− ε∇t∇t)−1

= ε(1l− ε∇t∇t)−1[∇s,∇t∇t](1l− ε∇t∇t)

−1.(102)

61

By Lemma C.3 and (101), we have

ε1/2∥∥(1l− ε∇t∇t)

−1[∇s,∇t∇t]ξ∥∥p

≤ 2ε1/2∥∥(1l− ε∇t∇t)

−1∇t[∇s,∇t]ξ∥∥p+ c1ε

1/2 ‖ξ‖p≤ 2κp ‖[∇s,∇t]ξ‖p + c1ε

1/2 ‖ξ‖p≤ c2 ‖ξ‖p .

(103)

Here we have used the L∞ bounds on ∂su, ∂tu, ∇t∂tu, and ∇t∂su. Now the fiverelevant terms are estimated as follows.

The term ‖ξ0‖p: By definition,

ξ0 = (1l− ε∇t∇t)−1(ξ − ε2∇tη).

Hence, by Lemma C.3,

‖ξ0‖p ≤ ‖ξ‖p + ε2 ‖∇tη‖p . (104)

The term ‖f0‖p: Consider the identity

(1l− ε∇t∇t)f0 − f + ε2∇tg

= ∇sξ0 − ε∇t∇t∇sξ0 −∇t∇tξ0 + ε∇t∇t∇t∇tξ0 −∇sξ + ε2∇t∇sη +∇t∇tξ

= ε2∇t∇t∇tη + ε2R(∂tu, ∂su)η + ε[∇s,∇t∇t]ξ0.

Apply the operator (1l−ε∇t∇t)−1 to this equation and use Lemma C.3 and (103)

to obtain

‖f0‖p ≤ ‖f‖p + κpε3/2 ‖g‖p + 2ε ‖∇tη‖p + ε2c3 ‖η‖p + ε1/2c2 ‖ξ0‖p , (105)

where c3 := ‖R‖∞ ‖∂su‖∞ ‖∂tu‖∞.

The term ε‖g0‖p: By (102), we have

g0 = ∇s∇tξ0

= (1l− ε∇t∇t)−1(∇t∇sξ + [∇s,∇t]ξ − ε2∇t∇t∇sη − ε2[∇s,∇t∇t]η

)

+ ε(1l− ε∇t∇t)−1[∇s,∇t∇t](1l− ε∇t∇t)

−1(∇tξ − ε2∇t∇tη

).

Hence, by Lemma C.3, (100), and (103),

ε ‖g0‖p ≤ κpε1/2 ‖∇sξ‖p + c3ε ‖ξ‖p + 2ε2 ‖∇sη‖p + c2ε

5/2 ‖η‖p+ c2ε

3/2∥∥(1l− ε∇t∇t)

−1(∇tξ − ε2∇t∇tη

)∥∥p

≤ κpε1/2 ‖∇sξ‖p + 2ε2 ‖∇sη‖p

+ ε(κpc2 + c3) ‖ξ‖p + 3c2ε5/2 ‖η‖p .

(106)

The term ε−1‖ξ1‖p: By (98), we have

ε−1ξ1 = ∇tη1 + ε∇tη −∇tη = ε∇tη −∇tη0.

62

Hence

ε−1‖ξ1‖p ≤ ε ‖∇tη‖p + ‖∇t∇tξ0‖p≤ ε ‖∇tη‖p + c0

(‖f0‖p + ‖ξ0‖p

).

(107)

In the last step we have used the parabolic estimate of Proposition C.2.

The term ‖η1‖p: By definition,

η1 = (1l− ε∇t∇t)−1 (

η −∇tξ + (ε2 − ε)∇t∇tη).

Hence, by the triangle inequality and Lemma C.3, we have

‖η1‖p ≤∥∥∥(1l− ε∇t∇t)

−1(η −∇tξ)

∥∥∥p+ ε

∥∥∥(1l− ε∇t∇t)−1∇t∇tη

∥∥∥p

≤ ‖η −∇tξ‖p + κp

√ε ‖∇tη‖p .

(108)

Insert the five estimates (104-108) into (99) to obtain (11), provided that ε issufficiently small. This proves Theorem 2.2.

The estimate for the inverse

Geometrically, the difference between the operatorsD0u andDε

u,v is the differencebetween configuration space and phase space, or between loops in M and loopsin T ∗M ∼= TM . Consider the embedding

LM → LTM : x 7→ (x, x).

The differential of this embedding is given by

Ω0(S1, x∗TM) → Ω0(S1, x∗TM ⊕ x∗TM) : ξ 7→ (ξ,∇tξ).

To compare the operators D0u and Dε

u := Dεu,∂tu

we must choose a projectiononto the image of this embedding (along u). At first glance it might seemnatural to choose the orthogonal projection with respect to the inner productdetermined by the (0, 2, ε)-Hilbert space structure. This is given by

(ξ, η) 7→ (1l− εα∇t∇t)−1(ξ − εβ∇tη)

with α = β = 2. Instead we introduce the projection operator

πε : Lp(S1, u∗TM)× Lp(S1, u∗TM) → W 1,p(S1, u∗TM)

given byπε(ξ, η) := (1l− ε∇t∇t)

−1(ξ − ε2∇tη). (109)

The reason for this choice becomes visible in the proof of Proposition C.5 be-low, which requires β = 2. Moreover, the estimates in Step 1 of the proof ofTheorem 2.3 are optimized for α = 1. We denote by ι : W 1,p(R×S1, u∗TM) →Lp(S1, u∗TM)× Lp(S1, u∗TM) the inclusion

ιξ0 := (ξ0,∇tξ0). (110)

The significance of these definitions lies in the next proposition and lemma. Theproofs rely on Lemma C.3.

63

Proposition C.5. Let u ∈ C∞(R × S1,M) be a smooth map such that thederivatives ∂su, ∂tu,∇t∂su,∇t∂tu,∇t∇t∂tu are bounded and define v := ∂tu.Then, for every p > 1, there exists a constant c > 0 such that

∥∥D0uπεζ − πεDε

uζ∥∥p≤ cε1/2 ‖ξ‖p + cε2 ‖η‖p + cε ‖∇tη‖p

for ε ∈ (0, 1] and compactly supported ζ = (ξ, η) ∈ Ω0(R× S1, u∗TM ⊕ u∗TM).The same estimate holds for (D0

u)∗πε − πε(Dε

u)∗. Moreover, the constant c is

invariant under s-shifts of u.

Lemma C.6. Let u ∈ C∞(R×S1,M), p > 1, κp as in (95) and ε ∈ (0, 1], then

‖ξ − πεζ‖p ≤ κpε1/2 ‖∇tξ − η‖p + ε ‖∇tη‖p

‖η −∇tπεζ‖p ≤ ‖∇tξ − η‖p + κpε1/2 ‖∇tη‖p

‖ζ − ιπεζ‖0,p,ε ≤ 2κpε1/2 ‖∇tξ − η‖p + 2κpε ‖∇tη‖p

‖πεζ‖p ≤ ‖ιπεζ‖0,p,ε ≤ 2κp ‖ζ‖0,p,εfor every compactly supported ζ = (ξ, η) ∈ Ω0(R× S1, u∗TM ⊕ u∗TM).

Proof. Denoteξ0 := πεζ = (1l− ε∇t∇t)

−1(ξ − ε2∇tη).

Then

ξ − ξ0 = ε(1l− ε∇t∇t)−1∇t(η −∇tξ) + (ε2 − ε)(1l− ε∇t∇t)

−1∇tη

and hence, by Lemma C.3,

‖ξ − ξ0‖p ≤ κpε1/2 ‖∇tξ − η‖p + ε ‖∇tη‖p

Similarly,

η −∇tξ0 = (1l− ε∇t∇t)−1(η −∇tξ) + (ε2 − ε)(1l− ε∇t∇t)

−1∇t∇tη

and hence, again by Lemma C.3,

ε ‖η −∇tξ0‖p ≤ ε ‖∇tξ − η‖p + κpε3/2 ‖∇tη‖p .

Take the sum of these two inequalities to obtain

‖ζ − ιπεζ‖0,p,ε ≤ ‖ξ − ξ0‖p + ε ‖η −∇tξ0‖p≤ 2κpε

1/2 ‖∇tξ − η‖p + 2κpε ‖∇tη‖pfor 0 < ε ≤ 1. Moreover, using Lemma C.3 the formula for ξ0 gives

‖ξ0‖p ≤ ‖ξ‖p + κpε3/2 ‖η‖p , ε ‖∇tξ0‖p ≤ κpε

1/2 ‖ξ‖p + 2ε2 ‖η‖p .Take these two inequalities to the power p and take the sum to obtain

‖ιπεζ‖p0,p,ε = ‖ξ0‖pp + εp ‖∇tξ0‖pp≤ (1 + κp

pεp/2) ‖ξ‖pp + (κp

pεp/2 + 2pεp)εp ‖η‖pp

≤ (2κp)p ‖ζ‖p0,p,ε

for 0 < ε ≤ 1. This proves Lemma C.6.

64

Proof of Proposition C.5. As above, denote

ξ0 := πεζ = (1l− ε∇t∇t)−1(ξ − ε2∇tη).

Then

D0uπεζ = ∇sξ0 −∇t∇tξ0 −R(ξ0, ∂tu)∂tu−∇ξ0∇Vt(u)

= (1l− ε∇t∇t)−1(∇sξ − ε2∇s∇tη −∇t∇tξ + ε2∇t∇t∇tη

)

+ ε(1l− ε∇t∇t)−1[∇s,∇t∇t]ξ0

+R((1l− ε∇t∇t)

−1ε2∇tη , ∂tu)∂tu+∇(1l−ε∇t∇t)−1ε2∇tη∇Vt(u)

−R((1l− ε∇t∇t)

−1ξ , ∂tu)∂tu−∇(1l−ε∇t∇t)−1ξ∇Vt(u).

Denote ζ′ := (ξ′, η′) := Dεuζ, then

πεDεuζ = (1l− ε∇t∇t)

−1(ξ′ − ε2∇tη′)

= (1l− ε∇t∇t)−1(∇sξ −R(ξ, ∂tu)∂tu−∇ξ∇Vt(u)

− ε2∇t∇sη − ε2∇t

(R(ξ, ∂su)∂tu

)−∇t∇tξ

).

Taking the difference we find

D0uπεζ − πεDε

uζ

= (1l− ε∇t∇t)−1(−ε2[∇s,∇t]η + ε2∇t∇t∇tη + ε2∇t

(R(ξ, ∂su)∂tu

))

+ ε(1l− ε∇t∇t)−1[∇s,∇t∇t]ξ0

+R((1l− ε∇t∇t)

−1ε2∇tη , ∂tu)∂tu+∇(1l−ε∇t∇t)−1ε2∇tη∇Vt(u)

+ (1l− ε∇t∇t)−1R(ξ, ∂tu)∂tu−R

((1l− ε∇t∇t)

−1ξ , ∂tu)∂tu

+ (1l− ε∇t∇t)−1∇ξ∇Vt(u)−∇(1l−ε∇t∇t)−1ξ∇Vt(u).

(111)

To finish the proof it remains to inspect the Lp norm of this expression line byline. Using Lemma C.3, we obtain for the first line

∥∥(1l− ε∇t∇t)−1(−ε2[∇s,∇t]η + ε2∇t∇t∇tη + ε2∇t

(R(ξ, ∂su)∂tu

))∥∥p

≤ ε2 ‖R‖∞ ‖∂su‖∞ ‖∂tu‖∞ ‖η‖p + 2ε ‖∇tη‖p+ κpε

3/2 ‖R‖∞ ‖∂su‖∞ ‖∂tu‖∞ ‖ξ‖p .(112)

Application of (103) with constant C1 := C results in an estimate for the secondline in (111), namely

∥∥ε(1l− ε∇t∇t)−1[∇s,∇t∇t]ξ0

∥∥p≤ ε1/2C1 ‖ξ‖p + ε5/2C1 ‖∇tη‖p . (113)

Lemma C.3 yields for line three in (111)

∥∥R((1l− ε∇t∇t)

−1ε2∇tη , ∂tu)∂tu+∇(1l−ε∇t∇t)−1ε2∇tη∇Vt(u)

∥∥p

≤(‖R‖∞ ‖∂tu‖2∞ + ‖∇∇V ‖∞

)ε2 ‖∇tη‖p .

(114)

65

Let us temporarily denoteT := 1l− ε∇t∇t.

Then the last two lines in (111) have the form [T−1,Φ] = T−1[Φ, T ]T−1 wherethe endomorphism Φ : u∗TM → u∗TM is given by Φξ = R(ξ, ∂tu)∂tu, respec-tively, Φξ = ∇ξ∇Vt(u). This term can be expressed in the form

[T−1,Φ]ξ = εT−1((∇t∇tΦ)T

−1ξ + 2(∇tΦ)T−1∇tξ

)

and hence ∥∥[T−1,Φ]ξ∥∥p≤ ε1/2κpC ‖ξ‖p .

Thus∥∥(1l− ε∇t∇t)

−1R(ξ, ∂tu)∂tu−R((1l− ε∇t∇t)

−1ξ, ∂tu)∂tu∥∥p

≤ ε1/2κpC2 ‖ξ‖p ,(115)

where C2 depends on ‖R‖C2 , ‖∂tu‖∞, ‖∇t∂tu‖∞, and ‖∇t∇t∂tu‖∞. Similarly,

∥∥(1l− ε∇t∇t)−1∇ξ∇Vt(u)−∇(1l−ε∇t∇t)−1ξ∇Vt(u)

∥∥p≤ ε1/2κpC3 ‖ξ‖p , (116)

where C3 depends on ‖V ‖C4 , ‖∂tu‖∞, and ‖∇t∂tu‖∞. The estimates (112-116)together give the desired Lp bound for (111) and this proves the first claim ofProposition C.5. The estimate for (D0

u)∗πεζ − (πεDε

u)∗ζ follows analoguously.

Since all constants appearing in the proof depend on L∞ norms of derivativesof u, they are invariant under s-shifts of u. This completes the proof of Propo-sition C.5.

The next lemma establishes the relevant estimates for the operator D0u and

its adjoint in the Morse–Smale case, i.e. when D0u is onto.

Lemma C.7. Assume SV is Morse-Smale and let u ∈ M0(x−, x+;V ). Then,for every p > 1, there is a constant c > 0 such that

‖η‖p + ‖∇sη‖p + ‖∇t∇tη‖p ≤ c∥∥(D0

u)∗η∥∥p

and‖ξ‖p + ‖∇sξ‖p + ‖∇t∇tξ‖p ≤ c

(∥∥ξ − (D0u)

∗η∥∥p+∥∥D0

uξ∥∥p

)

for all compactly supported vector fields ξ, η ∈ Ω0(R× S1, u∗TM).

Proof. By Theorem A.3, the operators D0u and (D0

u)∗ are Fredholm. Since SV

is Morse–Smale, the operator D0u is onto and (D0

u)∗ is injective. Moreover, the

operatorWp

u → Lpu ⊕ Lp

u/im (D0u)

∗ : ξ 7→ (D0uξ, [ξ])

is also an injective Fredholm operator. Hence the estimates follow from theopen mapping theorem.

66

Proof of Theorem 2.3. Fix a constant p > 1. Then the L∞ norms of ∂su, ∂tuand ∇t∂tu are finite by Theorem A.1 and ‖∇t∂su‖∞ is finite by Theorem A.2.Use the parabolic equations for u to conclude that ‖∇t∇t∂tu‖∞ is finite as well.Hence we are in a position to apply Theorem 2.2 and Proposition C.5. We provethe estimate in two steps.

Step 1. There are positive constants c1 = c1(p) and ε0 = ε0(p) such that

‖ζ‖0,p,ε ≤ ‖ξ‖p + ε1/2 ‖η‖p ≤ c1

(ε ‖(Dε

u)∗ζ‖0,p,ε + ‖πε(Dε

u)∗ζ‖p

)(117)

for every ε ∈ (0, ε0) and every compactly supported vector field ζ = (ξ, η) ∈Ω0(R× S1, u∗TM ⊕ u∗TM).

By Lemmata C.4 and C.7, there exists a constant c2 = c2(p) > 0 such that

‖ξ‖p + ‖∇sξ‖p + ‖∇tξ‖p + ‖∇t∇tξ‖p ≤ c2∥∥(D0

u)∗ξ∥∥p

(118)

for every compactly supported ξ ∈ Ω0(R× S1, u∗TM). Hence

‖ξ‖p ≤ ‖ξ − πεζ‖p + ‖πεζ‖p≤ ‖ξ − πεζ‖p + c2

∥∥(D0u)

∗πεζ∥∥p

≤ ‖ξ − πεζ‖p + c2∥∥(D0

u)∗πεζ − πε(Dε

u)∗ζ∥∥p+ c2 ‖πε(Dε

u)∗ζ‖p

≤ (κp + c2c3)ε(ε−1 ‖∇tξ − η‖p + ‖∇tη‖p

)+ c2 ‖πε(Dε

u)∗ζ‖p

+ c2c3

(ε1/2 ‖ξ‖p + ε2 ‖η‖p

)

≤ (κp + c2c3)c4ε ‖(Dεu)

∗ζ‖0,p,ε + c2 ‖πε(Dεu)

∗ζ‖p+ (c2c3 + κpc4 + c2c3c4)

(ε1/2 ‖ξ‖p + ε2 ‖η‖p

)

In the fourth step we have used Lemma C.6 and Proposition C.5 with a constantc3 = c3(p) > 0. The final step follows from Theorem 2.2 for the formal adjointoperator with a constant c4 = c4(p) > 0. Choose ε0 > 0 so small that

(c2c3 + κpc4 + c2c3c4)ε01/2 <

1

2. (119)

Then we can incorporate the term ‖ξ‖p into the left hand side and obtain

‖ξ‖p ≤ 2(κp + c2c3)c4ε ‖(Dεu)

∗ζ‖0,p,ε + 2c2 ‖πε(Dεu)

∗ζ‖p + ε3/2 ‖η‖p . (120)

Similarly,

‖η‖p ≤ ‖η −∇tπεζ‖p + ‖∇tπεζ‖p≤ ‖η −∇tπεζ‖p + c2

∥∥(D0u)

∗πεζ∥∥p

≤ (κp + c2c3ε1/2)c4ε

1/2 ‖(Dεu)

∗ζ‖0,p,ε + c2 ‖πε(Dεu)

∗ζ‖p+ (c2c3 + κpc4 + c2c3c4ε

1/2)(ε1/2 ‖ξ‖p + ε2 ‖η‖p

).

67

Use (119) again to obtain

‖η‖p ≤ 2(κp + c2c3)c4ε1/2 ‖(Dε

u)∗ζ‖0,p,ε + 2c2 ‖πε(Dε

u)∗ζ‖p + ‖ξ‖p . (121)

The assertion of Step 1 now follows from (121) and (120).

Step 2 We prove the theorem.

Let ε ∈ (0, ε0). By (92) for the formal adjoint operator (with a constant c5 > 0),we obtain

‖ζ‖1,p,ε ≤ c5ε2 ‖(Dε

u)∗ζ‖0,p,ε + c5 ‖ζ‖0,p,ε

≤ c5(ε2 + c1ε+ 2κpc1) ‖(Dε

u)∗ζ‖0,p,ε

(122)

Here we have also used the estimate (117) of Step 1 and Lemma C.6. It followsthat (Dε

u)∗ is injective and hence Dε

u is onto.

Let ζ = (ξ, η) ∈ Ω0(R × S1, u∗TM ⊕ u∗TM) be compactly supported anddenote

ζ∗ := (ξ∗, η∗) := (Dεu)

∗ζ.

Recall that c6 is the constant of Lemma C.7 and c3 is the constant of Proposi-tion C.5. By Lemma C.7, with ξ = πεζ

∗ and η = πεζ, we have

‖πεζ∗‖p ≤ c6

∥∥πεζ∗ − (D0

u)∗πεζ

∥∥p+ c6

∥∥D0uπεζ

∗∥∥p

≤ c6∥∥πε(Dε

u)∗ζ − (D0

u)∗πεζ

∥∥p+ c6

∥∥D0uπεζ

∗ − πεDεuζ

∗∥∥p

+ c6 ‖πεDεuζ

∗‖p≤ c3c6

(ε1/2 ‖ξ‖p + ε2 ‖η‖p + ε ‖∇tη‖p

)+ c6 ‖πεDε

uζ∗‖p

+ c3c6(ε1/2 ‖ξ∗‖p + ε2 ‖η∗‖p + ε ‖∇tη∗‖p

)

≤ 2c3c6(1 + c4ε1/2)ε1/2 ‖ζ‖0,p,ε + c6 ‖πεDε

uζ∗‖p

+ 3c3c6(1 + c4ε1/2)ε1/2 ‖ζ∗‖0,p,ε + c3c4c6ε ‖Dε

uζ∗‖0,p,ε

≤ c7ε1/2 ‖ζ∗‖0,p,ε + c3c4c6ε ‖Dε

uζ∗‖0,p,ε + c6 ‖πεDε

uζ∗‖p .

(123)

The fourth step follows by applying Theorem 2.2 twice, with the constant c4,namely for the operator (Dε

u)∗ to deal with the term ∇tη, and for the operator

Dεu to deal with the term ∇tη

∗. The final step follows from (122).Now it follows from Lemma C.6 that

‖ζ∗‖0,p,ε ≤ ‖ζ∗ − ιπεζ∗‖0,p,ε + ‖ιπεζ

∗‖0,p,ε≤ 2κpε

(ε−1 ‖∇tξ

∗ − η∗‖p + ‖∇tη∗‖p)+ ‖πεζ

∗‖p + ε ‖∇tπεζ∗‖p

≤ 2κpc4ε ‖Dεuζ

∗‖0,p,ε + ‖πεζ∗‖p + (2κp + 4κpc4)ε

1/2 ‖ζ∗‖0,p,ε≤ c4(2κp + c3c6)ε ‖Dε

uζ∗‖0,p,ε + (c7 + 2κp + 4κpc4)ε

1/2 ‖ζ∗‖0,p,ε+ c6 ‖πεDε

uζ∗‖p .

68

The third step follows from Theorem 2.2 for the operator Dεu and Lemma C.3.

The final step uses (123). Choosing ε0 > 0 sufficiently small, we obtain

‖ξ∗‖p ≤ ‖ζ∗‖0,p,ε ≤ 2c4(2κp + c3c6)ε ‖Dεuζ

∗‖0,p,ε + 2c6 ‖πεDεuζ

∗‖p . (124)

By (92), we have

‖ζ∗‖1,p,ε ≤ c5

(ε2 ‖Dε

uζ∗‖0,p,ε + ‖ζ∗‖0,p,ε

).

Combining this with (124) we obtain (13).We prove (12). By the triangle inequality and Lemmata C.6 and C.3, we

have

‖η∗‖p ≤ ‖η∗ −∇tπεζ∗‖0,p,ε +

∥∥∇t(1l− ε∇t∇t)−1(ξ∗ − ε2∇tη

∗)∥∥p

≤ κpε1/2(ε−1 ‖∇tξ

∗ − η∗‖p + ‖∇tη∗‖p)+ κpε

−1/2 ‖ξ∗‖p + 2ε ‖η∗‖p≤ κpc4ε

1/2 ‖Dεuζ

∗‖0,p,ε + 2κp(1 + c4ε)ε−1/2 ‖ζ∗‖0,p,ε

The last step follows from Theorem 2.2 for the operator Dεu. Similarly,

‖∇tξ∗‖p ≤ ‖∇tξ

∗ − η∗‖p + ‖η∗‖p≤ c5ε ‖Dε

uζ∗‖0,p,ε + c5ε ‖ξ∗‖p + (1 + c5ε

3) ‖η∗‖p .

Combining the last two estimates with (124) proves (12). Since all constants ap-pearing in the proof depend on L∞ norms of derivatives of u, they are invariantunder s-shifts of u. This proves Theorem 2.3.

Acknowledgement. Thanks to Katrin Wehrheim for pointing out to us thework of Marcinkiewicz and Mihlin and to Tom Ilmanen for providing the ideafor the proof of Lemma 4.5.

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