Pseudoholomorphic strips in symplectisations I : asymptotic behavior

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Pseudoholomorphic strips in symplectisations I:

Asymptotic behavior

Bandes pseudo-holomorphes en symplectisations I:

comportement asymptotique

Casim Abbas

Department of Mathematics, Wells Hall, Michigan State University, East Lansing, MI 48824, USA Received 21 November 2002; accepted 30 January 2003

Abstract

This paper is part of a larger program, the investigation of the Chord Problem in three dimensional contact geometry. The main tool will be pseudoholomorphic strips in the symplectisation of a three dimensional contact manifold with two totally real submanifoldsL₀, L₁as boundary conditions. The submanifoldsL₀andL₁do not intersect transversally. The subject of this paper is to study the asymptotic behavior of such pseudoholomorphic strips.

Résumé

Cet article fait partie d’un programme de travail plus grand : la recherche sur le problème de Chord en géometrie de contact en dimension trois. L’outil essentiel sont des bandes pseudo-holomorphes dans la symplectisation d’une variété contact à dimension trois avec la condition au bord suivante : les deux composantes de la frontière sont contenues dans deux sous-variétés totalement réellesL₀, L₁. Les variétésL₀etL₁ont des intersections non-transverses. Le sujet de cet article est l’étude du comportement asymptotique des solutions.

Keywords: Contact geometry; Pseudoholomorphic curves

1. Introduction

This paper is the first part of a larger program, the investigation of the chord problem in three dimensional contact geometry [4–6]. Let(M, λ) be a(2n+1)-dimensional contact manifold, i.e.λis a 1-form on M such thatλ∧(dλ)ⁿ is a volume form onM. The contact structure associated toλis the 2n-dimensional vector bundle

E-mail address: abbas@math.msu.edu (C. Abbas).

doi:10.1016/j.anihpc.2003.01.004

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ξ=kerλ→M, which is a symplectic vector bundle with symplectic structuredλ|ξ⊕ξ. There is a distinguished vector field associated to a contact form, the Reeb vector fieldXλ, which is defined by the equations

iX_λdλ≡0, iX_λλ≡1.

We denote byπ_λ:T M→ξ the projection along the Reeb vector field. The Chord Problem is about the global dynamics of the Reeb vector field. More precisely, the issue is the existence of so-called ‘characteristic chords’.

These are trajectoriesx of the Reeb vector field which hit a given Legendrian submanifoldLⁿ⊂(M, λ)at two different times t=0, T >0. We also ask forx(0) =x(T ), otherwise the chord would actually be a periodic orbit. Recall that a submanifoldL in a(2n+1)-dimensional contact manifold(M, λ)is called Legendrian if it is everywhere tangent to the hyperplane fieldξ and if it has dimensionn. We are mostly interested in the three- dimensional situation, the question is then whether a given Legendrian knot has a characteristic chord. The Chord problem should be viewed as the relative version of the Weinstein conjecture which deals with the existence of periodic orbits of the Reeb vector field.

Characteristic chords occur naturally in classical mechanics. In this context they are referred to as ‘brake–orbits’, and were investigated by Seifert in 1948 [20] and others since the 1970’s [7,9,21,22].

In 1986, V.I. Arnold conjectured the existence of characteristic chords on the three sphere for any contact form inducing the standard contact structure and for any Legendrian knot [8]. After a partial result by the author in [3] this conjecture was finally confirmed by K. Mohnke in [17]. It is natural to ask the existence question for characteristic chords not only forM=S³, but also for general contact manifolds. A new invariant for Legendrian knots and contact manifolds proposed by Y. Eliashberg, A. Givental and H. Hofer in [11] (‘Relative Contact Homology’) is actually based on counting characteristic chords and periodic orbits of the Reeb vector field.

The subject of the paper [6] is an existence result for characteristic chords which goes beyond the special classes of contact three manifolds investigated so far. The purpose of this paper and [4,5] is to establish a filling method by pseudoholomorphic curves where we use a surfaceF ⊂M=M³ with boundary, and where we start filling from a tangency at the boundary. Pseudoholomorphic curves are maps from a Riemann surface into an almost complex manifoldW satisfying a nonlinear Cauchy Riemann type equation. In our case, the manifoldW is the symplectisation(R×M, d(e^tλ))of the contact manifold(M, λ). We are going to consider a special type of almost complex structuresJ˜on R×M. We pick a complex structureJ:ξ→ξ such thatdλ◦(Id×J )is a bundle metric onξ. We then define an almost complex structure on R×M by demandingJ˜≡J onξ and sending∂/∂t (the generator of the R-component) onto the Reeb vector field. ThenJ (p)˜ has to mapX_λ(p)onto−∂/∂t.

IfSis a Riemann surface with complex structurej then we define a map

˜

u=(a, u):S→R×M to be a pseudoholomorphic curve if

Du(z)˜ ◦j (z)= ˜J

˜ u(z)

◦Du(z)˜ for allz∈S.

If(s, t)are conformal coordinates onSthen this becomes:

∂su˜+ ˜J (u)∂˜ tu˜=0.

We are interested only in pseudoholomorphic curves which have finite energy in the sense that E(u)˜ :=sup

φ∈Σ

S

˜

u^∗d(φλ) <+∞,

whereΣ:= {φ∈C^∞(R,[0,1])|φ0}. The Riemann surfaceSin this paper is an infinite stripS=R× [0,1], and we will impose a mixed boundary condition as follows: LetL⊂Mbe a homologically trivial Legendrian knot bounding an embedded surfaceD. A pointp∈Dis called singular ifTpD=kerλ(p). If the surface is oriented (by a volume formσ) and ifj:D&→M is the inclusion, then we define a vector fieldZ onDbyiZσ =j^∗λ.

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This vector field vanishes precisely in the singular points. The flow lines ofZdetermine a singular foliation of the surfaceDwhich does not depend on the particular choice of the volume form or the contact form. This singular foliation is also called the characteristic foliation ofD(induced by kerλ). Letp∈Dbe a singular point and denote byZ(p):T_pD→T_pDthe linearization of the vector fieldZinp. Letλ₁, λ₂be the eigenvalues ofZ(p). We say thatpis non-degenerate if none of the eigenvalues lie on the imaginary axis. A non-degenerate singular pointp is called elliptic ifλ₁λ₂>0 and hyperbolic ifλ₁λ₂<0. In the elliptic case the critical pointZ(p)=0 is either a source or a sink, and in the hyperbolic case it is a saddle point.

ChoosingDappropriately we may assume that there are only non-degenerate singular points, in particular there are only finitely many. We denote the surface without the singular points byD^∗. We consider the boundary value problem

˜

u=(a, u):S→R×M,

∂_su˜+ ˜J (u)∂˜ _tu˜=0,

˜

u(s,0)⊂R×L,

˜

u(s,1)⊂ {0} ×D^∗, 0< E (u) <˜ +∞.

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The subject of this paper is to investigate the behavior of solutionsu˜ for large|s|. The finiteness condition on the energy actually forces the solutions to converge to pointsp˜_±∈ {0} ×Lat an exponential rate. We first introduce suitable coordinates near the Legendrian knot, and we deformDnear its boundary, keepingL=∂Dfixed, in order to achieve a certain normal form forDnear its boundary. We then derive exponential decay estimates foru˜− ˜p_± and all its derivatives in coordinates. In local coordinates nearp˜_±the almost complex structureJ˜corresponds to some real(4×4)-matrix valued function which we denote byM. The main result of this paper is the following asymptotic formula

Theorem 1.1. For sufficiently large s₀ and ss₀ we have the following asymptotic formula for nonconstant solutionsvof (1) having finite energy:

v(s, t)=e

s s0α(τ ) dτ

e(t)+r(s, t)

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whereα:[s₀,∞)→R is a smooth function satisfyingα(s)→λ <0 ass→ ∞withλbeing an eigenvalue of the selfadjoint operator

A_∞:L²

[0,1],R⁴

⊃H_L^1,2

[0,1],R⁴

→L²

[0,1],R⁴ γ→ −M_∞γ ,˙ M_∞:= lim

s→∞M v(s, t)

.

Moreover,e(t)is an eigenvector ofA_∞belonging to the eigenvalueλwithe(t) =0 for allt∈ [0,1], andr is a smooth function so thatrand all its derivatives converge to zero uniformly intass→ ∞.

We will prove more about the decay of|λ−α(s)|,rand their derivatives:

Theorem 1.2. Letrandα(s)be as in Theorem 1.1. Then there is a constantδ >0 such that for each integerl0 and each multi-indexβ∈N²

sup

0t1

D^βr(s, t), d^l

ds^l

α(s)−λ

cβ,le⁻^δ^|^s^| with suitable constantsc_β,l>0.

The subscript ‘L’ inH_L^1,2([0,1],R⁴)indicates the boundary condition (see (21) for a precise definition). This formula is an essential ingredient for the rest of the program [4–6].

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The asymptotic behavior of holomorphic strips with mixed boundary conditions similar to ours was investigated in [19], but only for non-degenerate ends. We are dealing with a degenerate situation, i.e. the manifoldsL₀=R×L andL₁= {0} ×D do not intersect transversally. The degenerate situation is much more delicate: In the non- degenerate case the intersectionL₀∩L₁would consist of isolated points. Having shown that a pseudoholomorphic stripu(s, t)˜ with finite energy approachesL₀∩L₁as|s| → ∞one can fairly easily see that oscillations between two points inL₀∩L₁would cost too much energy, i.e. it would contradictE(u) <˜ ∞. In our case we have to show that the end of the solution cannot move along the 1–dimensional setL₀∩L₁while|s|grows. Analytically, degeneracy means that the operatorA_∞above has a nontrivial kernel. The strategy is to derive estimates for the

‘components’ ofu˜ orthogonal to the kernel ofA_∞(in a suitable sense). We will then show that they decay fast enough to force the component along the kernel ofA_∞to zero as well.

Degenerate ends were investigated in the paper [14], but only for pseudoholomorphic cylindersS=R×S¹ (periodic boundary condition int). Our problem requires a different approach. The paper [19] contains the decay estimate of Theorem 1.2 for the caseβ=0. Eduardo Mora proved Theorem 1.2 for pseudoholomorphic cylinders independently of the author in his Ph.D. thesis [18]. Because we are choosing specialJ˜andDnear{0}×Lsolutions to the boundary value problem (1) can be constructed explicitly near elliptic singular points on the boundary ofD (see [5]).

2. Simplifying the spanning surfaceDnear the boundary

In this section we will simplify the surfaceDnear its boundary to obtain a normal form in coordinates near the knotL=∂D. This is useful for the analysis later. In particular, we will be able to produce explicitly a family of finite energy strips coming out from elliptic singular points on the boundary.

If(M, λ)is a three dimensional contact manifold andLa Legendrian knot inMthen, by a well-known theorem of A. Weinstein (see [23,24,1]), there are open neighborhoodsU⊂Mof the knotL,V ⊂S¹×R²ofS¹× {(0,0)} and a diffeomorphismΨ:U→V, so thatΨ^∗(dy+x dθ )=λ|U, whereθ denotes the coordinate onS¹≈R/Z andx, yare coordinates on R². We will refer to this result as the ‘Legendrian neighborhood theorem’. If we are working near the knotLwe may assume that our contact manifold is(S¹×R², λ=dy+x dθ )and the knot is given byS¹×{(0,0)}. We will denote the piece of the spanning surfaceD∩Uagain byD. ChoosingUsufficiently small we may assume that all the singular points on the pieceD∩Ulie on the boundary, i.e.

p∈D∩U|T_pD=kerλ(p)

=

(θ_k,0,0)∈S¹×R²

1kN, N∈N.

We parameterizeDas follows:

D=

θ , x(θ , r), y(θ , r)

∈S¹×R²|r, θ∈ [0,1] ,

wherex, yare suitable smooth functions which are 1-periodic inθand satisfy x(θ ,0)≡y(θ ,0)≡0.

Moreover we orientDin such a way that the above parameterization([0,1] × [0,1], dθ∧dr)→Dis orientation preserving. We orientLbyv=d/dθ, so that(v, ν) is positively oriented, whereν denotes the inward normal vector. A point(θ0,0,0)is a singular point if and only if∂ry(θ0,0)=0. Since also∂θy(θ0,0)=∂θx(θ0,0)=0 and D is embedded, we conclude that ∂rx(θ0,0) =0. The tangent space T_(θ₀_,0,0)D is oriented by the basis (∂/∂θ , ∂rx(θ0,0)∂/∂x). On the other hand, the contact structure kerλ(θ0,0,0)is oriented by(∂/∂θ ,−∂/∂x). The singular point(θ0,0,0)is called positive if these two orientations coincide, which is the case for∂rx(θ0,0) <0, otherwise(θ0,0,0)is called negative. Hence in the case of a positive (negative) singularity, the surfaceDlies on the side of the negative (positive)x-axis. We would like to perturbDnear its boundary, leaving the boundary fixed, so that the number and type of the singularities does not change and the new surface has some kind of normal form

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near its boundary. the following is the main result of this section. It is an immediate consequence of Propositions 2.2 and 2.3 below.

Proposition 2.1. Let(M, λ)be a three-dimensional contact manifold. Further, letLbe a Legendrian knot andD an embedded surface with∂D=Lso that all the singular points are non-degenerate. We denote the finitely many singular points on the boundary byek, 1kN (ordered by moving in the direction of the orientation ofL).

Then there is an embedded surfaceDhaving the same boundary asDwhich differs fromDonly by aC⁰-small perturbation supported nearLhaving the same singular points asDso that the following holds:

There is a neighborhoodU ofLand a diffeomorphismΦ:U→S¹×R²so that

• Φ^∗(dy+x dθ )=λ|U, (θ , x, y)∈S¹×R²,

• Φ(L)=S¹× {(0,0)},

• Φ(ek)=(θk,0,0), 0θ₁<· · ·< θN<1,

• Φ(U∩D)= {(θ , a(θ )r, b(θ )r)∈S¹×R²|θ , r∈ [0,1]}, wherea, bare smooth 1-periodic functions with:

• b(θk)=0 andb(θ )is nonzero ifθ =θk,

• a(θk) <0 ifekis a positive singular point,a(θk) >0 ifek is a negative singular point,

• ifekis elliptic then−1< b(θk)/a(θk) <0,

• ife_kis hyperbolic then the quotient ^b_a(θ^(θ^k⁾

k) is either strictly smaller than−1 or positive,

• ahas exactly one zero in each of the intervals[θ_k, θ_k₊₁],k=1, . . . , N−1 and[θ_N,1] ∪ [0, θ₁],

• Ife_kis an elliptic singular point and if|θ−θ_k|is sufficiently small then we haveb(θ )= −¹₂a(θ )(θ−θ_k).

We consider first the situation near boundary singular points.

2.1. A normal form for the spanning surface near boundary singularities

We first simplify the surfaceDnear singular points on the boundary:

Proposition 2.2. Let L be a Legendrian knot in a three dimensional contact manifold(M, λ) and let D⊂M be an embedded surface with∂D=L. Assume that the singular points of the characteristic foliation onDare nondegenerate. Denote by(θ , x, y)∈S¹×R the coordinates nearLprovided by the Legendrian neighborhood theorem. IfDis parameterized by

θ , x(θ , r), y(θ , r)

∈S¹×R|r∈ [0,1]

nearLthen there is an embedded surfaceDwith the following properties:

• D is obtained fromDby aC⁰-small perturbation supported near the boundary singular points leaving the boundary fixed, i.e.∂D=∂D=L.

• Dhas the same singular points asD.

• If(θ0,0,0)is a boundary singularity and D=

θ , x(θ , r), y(θ , r)

∈S¹×R|r∈ [0,1] then

y(θ , r)=cx(θ , r)(θ−θ₀)+b

2x(θ , r)²,

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where

1. c= −¹₂ andb=0 if(θ₀,0,0)is elliptic,

2. c∈(−∞,−1)∪(0,+∞)if(θ₀,0,0)is hyperbolic.

Proof. Let us first point out how to recognize the type of the singularity(θ₀,0,0)in the above parameterization.

Since the Jacobian of the mapΨ (θ , r)=(θ , x(θ , r))at the point(θ0,0)has rank 2, there is a local inverse and we parameterizeDby

D= θ , x,

y◦Ψ⁻¹ (θ , r)

,

where(θ , x)is sufficiently near to(θ₀,0). Note that in the case of a positive (negative) singular point(θ₀,0,0)the mapΨ⁻¹is only defined for non-positive (non-negative)x. We write

f (θ , x):=

y◦Ψ⁻¹ (θ , x) and note that

• f (θ ,0)≡0,

• ∂xf (θ0,0)=0 since(θ0,0,0)is a singular point.

We may extendf smoothly so that it is defined for small|x|regardless of the sign ofx. Write f (θ , x)=a

2(θ−θ₀)²+b

2x²+c(θ−θ₀)x+h(θ , x)

witha=∂θ θf (θ0,0),b=∂xxf (θ0,0),c=∂xθf (θ0,0), andhof order at least 3 in(θ−θ0, x). Note thata=0 and alsoh(θ ,0)=0, hence

f (θ , x)=b

2x²+c(θ−θ₀)x+h(θ , x).

Investigate now the admissible values for the constantsbandc. The surfaceDis given byH⁻¹(0), where H (θ , x, y):=y−f (θ , x).

Then the vector fieldV_H, which is defined byi_V

Hdλ=(i_X_λdH ) dλ−dH andi_V

Hλ=0, is given by V_H(θ , x, y)= −∂_xf (θ , x) ∂

∂θ +

x+∂_θf (θ , x) ∂

∂x+x∂_xf (θ , x)∂

∂y,

and it induces the characteristic foliation onD. Its linearizationV_H(θ , x, y)is given by



−∂_xθf (θ , x) −∂_xxf (θ , x) 0

∂_{θ θ}f (θ , x) 1+∂_xθf (θ , x) 0 x∂_xθf (θ , x) ∂_xf (θ , x)+x∂_xxf (θ , x) 0



.

The contact structure kerλ(θ₀,0,0)is generated by the vectors(1,0,0)and(0,1,0). We representV_H (θ₀,0,0)by the matrix

−∂_xθf (θ₀,0) −∂_xxf (θ₀,0) 0 1+∂xθf (θ0,0)

=

−c −b

0 1+c

.

The singular point(θ₀,0,0)is then hyperbolic ifc(c+1) >0 and elliptic ifc(c+1) <0. Let us translate this into our original parameterization(θ , x(θ , r), y(θ , r))ofD. Usingf ◦Ψ =y, we compute

∂θ ry(θ0,0)=c ∂rx(θ0,0),

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so that we are in the following situation: A singular point(θ₀,0,0)is 1. positive if∂rx(θ0,0) <0,

2. negative if∂_rx(θ₀,0) >0,

3. elliptic if(∂_{θ r}y/∂_rx)(θ₀,0)∈(−1,0), and

4. hyperbolic if(∂_{θ r}y/∂_rx)(θ₀,0)∈(−∞,−1)∪(0,+∞).

We will remove now the higher order termhby a perturbation. Take a smooth functionβ:[0,∞)→ [0,1]with β≡0 on[0,1],β≡1 on[2,∞)andβ0. For smallδ >0 we defineβδ:=β((θ²+x²)/δ²)and

f_δ(θ , x):=b

2x²+cx(θ−θ₀)+β_δ(θ−θ₀, x)h(θ , x).

This perturbation takes place in a small neighborhood of the singular point(θ0,0,0). We have to show that the new surface given by the graph offδhas the same singularities asDprovidedδ >0 was chosen sufficiently small.

We proceed indirectly. Assume that for any sequenceδ_n 0 there is a singular point(θ_n, x_n, f_δ_n(θ_n, x_n))on the surfaceDδn given by the graph of f_δ_n which satisfies (θ_n−θ₀)²+x_n²2δ_n²and is different from (θ₀,0,0). If (θ_n, x_n, f_δ_n(θ_n, x_n))is singular then

0=∂_θf_δ_n(θ_n, x_n)+x_n

=(c+1)xn+βδ_n(θn−θ₀, xn)∂θh(θn, xn)+2(θ_n−θ₀) δ_n² β

(θ_n−θ₀)²+x_n² δ²_n

h(θn, xn) and

0=∂_xf_δ_n(θ_n, x_n)

=bxn+c(θn−θ0)+βδ_n(θn−θ0, xn)∂xh(θn, xn)+2x_n δ²_n β

(θ_n−θ₀)²+x²_n δ²_n

h(θn, xn).

We write shortly

0=(c+1)x_n+β_δ_n∂_θh+2(θ_n−θ₀) δ²_n β_δ

nh, 0=bx_n+c(θ_n−θ₀)+β_δ_n∂_xh+2x_n δ²_n β_δ

nh. (3)

Remark. The reader should be aware thatβ_δ

nis not the derivative ofβ_δ_n, but the rescaled derivative ofβ β_δ

n:=β

(θn−θ0)²+x_n² δ_n²

. Eq. (3) is the same as

x_n

θ_n−θ₀

= 1

c(c+1)

c 0

−b c+1

H (θ_n, x_n) (4)

with

H (θ_n, x_n)= −

βδn∂θh+^2(θⁿ_δ⁻2^θ⁰⁾ n β_δ

nh βδ_n∂xh+^2x_δ2ⁿ

n

β_δ

nh

, which satisfies

H (θ_n, x_n)

(θn−θ0)²+x_n² →0

asn→ ∞sinceH is of order at least 2 in(θ_n−θ₀, x_n). Dividing Eq. (4) by

(θ_n−θ₀)²+x_n²and passing to the limitn→ ∞we obtain a contradiction.

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Hence we may assume thatDis given by the graph of f (θ , x)=b

2x²+cx(θ−θ₀)

forx²+(θ−θ₀)²sufficiently small. Consider now the case where(θ₀,0,0)is an elliptic singularity. Take the same smooth functionβ as before and define forδ >0

f_δ(θ , x):=β

(θ−θ0)²+x² δ²

b

2x²+cx(θ−θ₀),

so that f_δ ≡f if (θ −θ₀)²+x²2δ² and f_δ(θ , x)=cx(θ−θ₀) if (θ −θ₀)²+x²δ². Writing β_δ:=

β(((θ−θ0)²+x²)/δ²),β_δ:=β(((θ−θ0)²+x²)/δ²)as before, the condition of(θ , x, fδ(θ , x))being a singular point is

0=

c bβ_δ+^bx_δ2²β_δ

bx²

δ² β_δ c+1

θ−θ0

x

which implies

0=c(c+1)−b²

x²

δ²β_δβ_δ+x⁴ δ⁴(β_δ)²

c(c+1)

in contradiction to the fact that(θ₀,0,0)is an elliptic singularity. Hence we may assume that f (θ , x)=cx(θ−θ0)

near an elliptic singularity (θ0,0,0), where c∈(−1,0). Now we will carry out a last modification to achieve c= −¹₂. We take a smooth function

β: R→

min{c,−1/2},max{c,−1/2}

⊂(−1,0)

withβ(s)= −¹₂for|s|1 andβ(s)=cfor|s|2. Define for smallδ >0 fδ(θ , x):=β

(θ−θ₀)²+x² δ²

x(θ−θ0).

Again, we did not create any new singular points. This completes the proof of Proposition 2.2. ✷ 2.2. Perturbing the spanning surface near the Legendrian knot

We will now show the following:

Proposition 2.3. LetL be a Legendrian knot in a three dimensional contact manifold(M, λ)and letD be an embedded surface with∂D=L. Assume that the singular points onDare non-degenerate.

Then there is an embedded surfaceDwith∂D=Lwhich differs fromDby aC⁰-small perturbation supported nearLand leavingLfixed so thatDandDhave the same singular points and the following holds:

There is a neighborhoodU ofLand a diffeomorphismΦ:U→S¹×R²so that 1. Φ^∗(dy+x dθ )=λ|U,(θ , x, y)∈S¹×R,

2. Φ(L)=S¹× {(0,0)},

3. Φ(U∩D)= {(θ , a(θ )r, b(θ )r)∈S¹×R|θ∈S¹≈R/Z, r∈ [0,1]}, where θ→

a(θ ) b(θ )

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is a smooth closed curve in R²\{0}with the following properties:

(a) tb(L)=deg

θ→ a(θ )

b(θ )

,

where tb denotes denotes the Thurston–Bennequin invariant of the Legendrian knot (see [10]).

(b) b(θ₀)=0 if and only ifΦ⁻¹(θ₀,0,0)is a singular point onL, (c) a singular pointΦ⁻¹(θ₀,0,0)is

(i) positive (negative) ifa(θ₀) <0(a(θ₀) >0), (ii) elliptic ifc=b(θ₀)/a(θ₀)∈(−1,0),

(iii) hyperbolic ifc=b(θ₀)/a(θ₀)∈(−∞,−1)∪(0,+∞),

(d) forθnearθ₀, whereΦ⁻¹(θ₀,0,0)is a singular point, we haveb(θ )=c(θ−θ₀)a(θ ),

(e) ifΦ⁻¹(θ₀,0,0)andΦ⁻¹(θ₁,0,0)are singular points of opposite sign withθ₀< θ₁, so that all the points (θ ,0,0)are non-singular forθ∈(θ₀, θ₁), thenahas exactly one zero in the interval(θ₀, θ₁).

Proof. We parameterizeDagain by D=

θ , x(θ , r), y(θ , r)

∈S¹×R²|θ∈S¹, r∈ [0,1] and we expandx, yas follows:

x(θ , r)=∂rx(θ ,0)r+h(θ , r), y(θ , r)=∂_ry(θ ,0)r+k(θ , r),

whereh, kare of order at least 2 inrand 1-periodic inθ. For smallrand|θ−θ0|, where(θ0,0,0)is a boundary singular point, we have

y(θ , r)=cx(θ , r)(θ−θ₀)+b

2x²(θ , r) (5)

by Proposition 2.2. In the case of an elliptic singularity we may assume that c= −¹₂ andb=0. We want to perturbD near its boundary leaving ∂Dfixed, so that the higher order termsh andk disappear. We will only indicate the necessary steps and leave the details to the reader. The verification that no new singularities are created is completely straight forward using the normal form (5) near the singular points. Pick a smooth function β:[0,∞)→ [0,1]withβ≡0 on[0,1],β≡1 on[2,∞)and 0β(s)2 for alls0. We define

x_δ(θ , r):=∂_rx(θ ,0)r+β

r

δ

h(θ , r), y_δ(θ , r):=∂_ry(θ ,0)r+β

r

δ

k(θ , r) and

Dδ=

θ , x_δ(θ , r), y_δ(θ , r)

∈S¹×R²|θ∈S¹, r∈ [0,1] .

Forr2δthe perturbed surfaceDδcoincides withDand we have∂Dδ=∂D=L. The surfaceDδhas the same singularities on the boundary asDand thatDδ has no singularities in the range 0< r <2δ providedδ >0 was chosen sufficiently small. It remains to verify that the surfaceDδ is embedded for sufficiently smallδ. If it were not then we could find sequencesδk 0, 0rk, r_k 2δkandθksuch that

∂_rx_δ_k(θ_k, r_k), ∂_ry_δ_k(θ_k, r_k)

=(0,0) for allk(surface not immersed) or

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xδ_k(θk, rk)=xδ_k(θk, r_k), yδ_k(θk, rk)=yδ_k(θk, r_k)

for allk(surface has self-intersections). Both assertions contradict the fact that(∂rx(θ ,0), ∂ry(θ ,0)) =(0,0)for allθ, and can therefore not occur.

Hence we may assume that nearLwe have D=

(θ , a(θ )r, b(θ )r)∈S¹×R²|r∈ [0,1] , where the map

θ→ a(θ )

b(θ )

is a closed curve in R²\{0}. A point(θ₀,0,0)is a singular point if and only ifb(θ₀)=0 and it is 1. positive ifa(θ₀) <0,

2. negative ifa(θ0) >0,

3. elliptic ifb(θ0)/a(θ0)∈(−1,0)and

4. hyperbolic ifb(θ0)/a(θ0)∈(−∞,−1)∪(0,+∞).

Ifrandθ−θ0are sufficiently small, where(θ0,0,0)is a singular point, then we compute with the normal form (5):

b(θ )=∂_ry(θ ,0)=c∂_rx(θ ,0)(θ−θ₀)=ca(θ )(θ−θ₀), (6) so if we use the parameters(θ , ρ=a(θ )r)instead of(θ , r)thenDis given by

D=

(θ , ρ, cρ(θ−θ₀))

near(θ₀,0,0).

Hence the modification that we carried out on Din this section did not affect the normal form near boundary singularities that we have constructed in the previous section.

In this picture it is easy to understand the Thurston–Bennequin invariant of the knotL. Let us shiftLalong the Reeb vector field to get a knot

L:=

(θ ,0, δ)∈S¹×R²|θ∈ [0,1]

with some small constantδ. ThenLandDintersect if and only if a(θ )=0 and r= δ

b(θ ).

The conditiona(θ )=0 means that the Reeb vector fieldX_λis tangent toDat the point(θ ,0, δ). Without affecting the value of the intersection number int(L,D)we may perturb the loop(a(θ ), b(θ ))slightly so thata(θ ) =0 whenevera(θ )=0. Then we compute withλ=dy+x dθand

S:=

θ∈ [0,1] |a(θ )=0 and signb(θ )=signδ :

tb(L)=

θ∈S

sign



(λ∧dλ)_(θ,0,δ)







 1 a(θ )δ/b(θ ) b(θ )δ/b(θ )



,

0

0 b(θ )

,

1

0 0









=

θ∈S

sign

−δa(θ )

=deg

θ→ a(θ )

b(θ )

. (7)

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Assume now that (θ₀,0,0) and(θ₁,0,0)are singularities of opposite sign with θ₀< θ₁, so that all the points (θ ,0,0)withθ∈(θ₀, θ₁)are not singular. Let us assume that(θ₀,0,0)is the negative singularity. Then

• b(θ₀)=b(θ₁)=0 andbis nonzero on(θ , θ₁).

• a(θ0) >0, a(θ1) <0.

We would like to perturbDnearL, leaving the boundary fixed, so thatahas only one zero in the interval(θ0, θ1).

Letδ >0 and pick a smooth functionβso thatβ≡0 on[0, δ]andβ≡1 on[2δ,∞). Letaˆbe a 1-periodic function which coincides witha except on some interval[θ₀+ε, θ₁−ε], and which has exactly one zero betweenθ₀and θ₁. We define

˜

a(θ , r):=

1−β(r) ˆ

a(θ )+β(r)a(θ ) and denote the new surface by

Dδ:=

(θ ,a(θ , r)r, b(θ )r)˜ ,

which has the same number and type of singularities as D because we did not change the function b and because a˜ coincides with a near θ₀ and θ₁. Moreover, Dδ is embedded since it is immersed and the map (θ , r)→(θ ,a(θ , r), b(θ ))˜ is also injective. This completes the proof of Proposition 2.3. ✷

Remark. The negative singularities correspond to the points where the curveθ−→^γ a(θ )+ib(θ )∈C\{0}hits the positive real axis. Similarly, positive singularities corresponds to the intersection ofγ with the negative real axis.

2.3. The non-Lagrangian part of the boundary condition

The submanifold R×Lis a Lagrangian submanifold of the symplectisation(R×M, d(e^tλ)). However, the submanifold{0}×Dis only totally real with respect to anyJ˜away from the singular points. These two submanifold serve as boundary conditions for our boundary value problem, and we have to find a way to deal with{0} ×Din order to derive apriori estimates. The problem is the following: The fibers of the vector bundlesT (R×L)and J T (R˜ ×L)are orthogonal with respect to theJ˜-invariant metric g=d(e^tλ)◦(J˜×Id)whileT ({0} ×D)and J T (˜ {0} ×D)are only transverse, but not orthogonal. On the other hand, we will need this orthogonality to prove asymptotic decay estimates later (without it certain operators would fail to be self-adjoint). The way out is the following: Instead of using the metricg above, we use a different one where we have orthogonality. We will be able to control this metric if we do estimates later on. There is a 2-formωnear the intersection set{0} ×Lof R×L and{0} ×Dwhich is nondegenerate away from the singular points so that both submanifolds become Lagrangian with respect toω, andωis compatible with the almost complex structureJ˜. In general, we cannot expectωto be closed, unless we weaken our requirements and replace compatibility by tameness (i.e.ω(v,J v) >˜ 0 for allv =0).

It will turn out that we need the compatibility condition, but we do not needωto be closed. We construct such a 2-form explicitly in local coordinates. We will confine ourselves to a special almost complex structureJ˜near {0} ×Lwhich will also be used in the subsequent papers [4–6].

From now on we pick an almost complex structureJ˜on R×M, where the correspondingJ:ξ→ξ has the following form in local coordinates near{0} ×L:

J (θ , x, y)·(1,0,−x):=(0,−1,0), J (θ , x, y)·(0,1,0):=(1,0,−x). (8) Lemma 2.4. If Uk ⊂M are disjoint open neighborhoods of the singular points ek, k =1, . . . , N, on the boundaryL=∂Dthen there exist an open neighborhoodV ⊂M ofLand a nondegenerate 2-form ωdefined onW=R×(V\ ∪kU_k)⊂R×M, so thatω|T ({0}×D)≡0,ω|T (R×L)≡0 and the formωis compatible withJ˜, i.e.ω◦(Id× ˜J )is a Riemannian metric.

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Proof. Use the coordinates(θ , x, y)∈R³nearLwhich we derived in Section 2, where the contact form equals dy+x dθand{0} ×Dis represented by

0, θ , a(θ )r, b(θ )r

∈ {0} ×R³|r, θ∈ [0,1] .

Denoting the standard Euclidean product on R⁴ by #·,·$, we have to find a function with values in the set of skew-symmetric 4×4-matricesΩ(τ, θ , x, y)such that

1. #·, ΩJ .˜$is a metric,

2. #v, Ωw$|(τ,θ,0,0)=0 for allv, w∈T_(τ,θ,0,0)(R×L),

3. #v, Ωw$|(0,θ,q(θ )y,y)=0 for allv, w∈T(0,θ,q(θ )y,y)({0} ×D), whereq(θ ):=a(θ )/b(θ ).

The matrix ofJ˜is given by

J (τ, θ , x, y)˜ =





0 −x 0 −1

0 0 1 0

0 −1 0 0

1 0 −x 0



.

We writeΩ=(ω_kl)₁_k,l₄withω_kl= −ω_lk. If we choose

ω(τ, θ , x, y)= −xC dτ∧dθ−q(θ )dτ∧dx+C dτ∧dy−dθ∧dx+q(θ ) dθ∧dy andω= #·, Ω.$, where

C >max

0,supq²(θ ) ,

then the matrixΩJ˜is positive definite ifx, yare sufficiently small. ✷

3. Asymptotic behavior at infinity

Assume we have a solution of:











˜

u=(a, u):S→R×M,

∂_su˜+ ˜J (u)∂˜ _tu˜=0,

˜

u(s,0)⊂R×L,

˜

u(s,1)⊂ {0} ×D^∗∗, E(u) <˜ +∞,

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whereS:=R× [0,1]andD^∗∗is the spanning surfaceDwithout some open neighborhoodUof the set of singular pointsΓ. We will show that the condition of finite energy forces the solution to converge to points on the knotL for|s| → ∞, more precisely

˜

u(s, t)→ ˜p_±∈ {0} ×L

ass→ ±∞uniformly int. We will also show that this convergence is of exponential nature. This fact will be crucial for the nonlinear Fredholm theory in [4].

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3.1. The solutions approach the Legendrian asymptotically

As a first step, we will show that the ends of a finite energy stripu˜ have to approach the knot{0} ×L⊂R×M asymptotically. This actually works under the weaker assumptionu(s,˜ 1)∈ {0} ×D. The main result of this section is Proposition 3.4 below.

Lemma 3.1. Assumeu˜:S→R×Msatisfies Eq. (9) above. If in addition

S

u^∗dλ=0 thenu˜must be constant.

Proof. The mapu˜=(a, u)satisfies the following system of equations:

πλ∂su+J (u)πλ∂tu=0,

∂_sa−λ(u)∂_tu=0,

∂_ta+λ(u)∂_su=0.

Since

S

u^∗dλ=

S

dλ(π_λ∂_su, π_λ∂_tu) ds∧dt

=1 2

S

|πλ∂su|²J+ |πλ∂tu|²J

ds∧dt

=0,

where| · |²_J=dλ(·, J.), we conclude thatπ_λ∂_su=π_λ∂_tu≡0 and therefore

%a ds∧dt= −d(da◦i)=u^∗dλ=0,

hencea:S→R is harmonic and satisfiesa(s,1)≡0. Because ofu(s,0)∈Lwe also have

∂ta(s,0)= −λ u(s,0)

∂su(s,0)≡0.

Define nowf:S→R by f (s, t):=

t

0

∂_sa(s, τ ) dτ so that∂_tf=∂_saand

∂sf (s, t)= − t

0

∂t ta(s, τ ) dτ= −∂ta(s, t)+∂ta(s,0)= −∂ta(s, t).

ThenΦ:=a+if:S→C is holomorphic and satisfies Φ(s,0)∈R, Φ(s,1)∈iR.

Case 1.|∇Φ|is bounded. We define Φ:S:=R× [−1,+1] →C

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by

Φ(s, t) :=

!Φ(s, t) ift0, Φ(s,−t) ift <0.

Note thatΦis holomorphic. Let bˆ:=∂_s(ReΦ ) :S→R.

Thenbˆis harmonic,C:=sup_S| ˆb|<+∞by assumption andb(s,ˆ ±1)≡0. Defining ˆ

c(s, t):=

t

0

∂_sb(s, τ ) dτˆ − s

0

∂_tb(σ,0) dσ,ˆ

we compute∂_tcˆ=∂_sbˆ and∂_scˆ= −∂_tb, henceˆ δ:= ˆb+icˆ:S→C is holomorphic with bounded real part. The functiong:=e^δis also holomorphic and satisfies

|g|e^C, g(s,±1)=1.

Letε >0 and define a holomorphic function onSby h_ε(z):= 1

1−iε(z+i). We compute withz=s+it

hε(z)²= 1

(1+ε(1+t))²+ε²s²1.

Fors =0 we have|hε(z)|²1/(ε²s²), hence the holomorphic functionghεsatisfies g(z)hε(z)1

wheneverz∈∂Ω, where Ω:=

−ε⁻¹e^C, ε⁻¹e^C

× [−1,+1].

Using the maximum principle, we conclude that|gh_ε|1 on all ofΩ, but outsideΩ we also have g(z)hε(z) e^C

ε|s|1.

Keepingz∈Sfixed and passing to the limitε 0 we conclude that|g(z)| =e^b(z)^ˆ 1, henceb(z)ˆ 0. Repeating the same argument with−δ instead ofδ, we also obtain− ˆb(z)0, hence∂_s(ReΦ ) = ˆb(z)≡0. We know now that ReΦis harmonic, does not depend onsand satisfies ReΦ(s, ±1)≡0. This implies that ReΦis identically zero and therefore alsoa≡0. In view of

∂su=πλ∂su+

λ(u)∂su Xλ(u) and

∂_tu=π_λ∂_tu+

λ(u)∂_tu X_λ(u) we conclude thatumust be constant.

Case 2.|∇Φ|is unbounded. Pick sequencesz_k∈S,ε_k 0 so that ε_k∇Φ(z_k)→ +∞.

By a lemma of Hofer (see [15], Chapter 6.4, Lemma 5 and [1]) we find sequencesz_k=s_k+it_k∈S,ε_k 0 so that

• εkRk:=εk|∇Φ(zk)| → +∞,

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• |zk−z_k|ε_k,

• |∇Φ(z)|2R_kwhenever|z−z_k|ε_k.

We may assume without loss of generality thattk→t₀∈ [0,1]. We consider the following cases after choosing a suitable subsequence:

1. −tkRk→ −∞,

(a) Rk(1−tk)→l∈ [0,+∞), (b) Rk(1−tk)→ +∞,

2. −tkRk→ −l∈(−∞,0], thenRk(1−tk)→ +∞. Let us begin with the case 1(b). We define

Ω_k:=R×

−t_kR_k, R_k(1−t_k) and the holomorphic mapsΦ_k:Ω_k→C by

Φk(z):=Φ

zk+zR⁻_k¹

−Φ(zk) so that

∇Φk(0)=1, Φk(0)=0

and ∇Φk(z)2

ifz∈B_ε_k_R_k(0)∩Ω_k. Using the Cauchy integral formula for higher derivatives we find for each compact subsetK of C a numberk₀so thatK⊂B_ε_k_R_k(0)∩Ω_k for allkk₀and all the mapsΦ_k are bounded inC^∞(K)uniformly inkk₀. By the Ascoli–Arzela theorem, some subsequence of(Φ_k)converges inC_loc^∞ to an entire holomorphic functionΨ satisfying

∇Ψ (z)2, ∇Ψ (0)=1 and Ψ (0)=0.

By Liouville’s theoremΨ must be an affine function. Letφ∈Σand defineφ_k∈Σby φ_k(s):=φ

s−ReΦ(z_k) and

τ_φ(s, t):=φ(s) ds∧dt.

We estimate usingu^∗dλ=0:

Ω_k

Φ_k^∗τ_φ=

S

Φ^∗τ_φ_k=

S

φ_k(a)Φ^∗(ds∧dt)

=

S

φ_k(a) da∧u^∗λ=

S

˜

u^∗d(φ_kλ)E(u).˜

For every compactK⊂C we have

K

Φ_k^∗τ_φ^k−→^→∞

K

Ψ^∗τ_φ.