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Extremal discs in almost complex spaces

HERVE´ GAUSSIER ANDJAE-CHEONJOO

Abstract. We give a necessary condition for the existence of extremal pseu- doholomorphic discs on some domains in(R2n,J)where J is a small almost complex deformation of the standard complex structure.

Mathematics Subject Classification (2010):32F45 (primary); 32G05, 32H99, 53C15 (secondary).

1.

Introduction

This paper deals with the study of extremal discs in almost complex manifolds. In the seminal paper [14] L. Lempert proved that every convex domain admits a singular foliation by such discs which are proper and smooth up to the boundary in case the domain = {ρ < 0}is smooth. The proof of existence of the foliation relies on an analytic characterization of extremal discs: if f is an extremal disc in then f is stationary, meaning that there exists a positive function p defined on the unit circlesuch that the functionζζp(ζ )dρ(f)admits a holomorphic extension to the unit disc. Important applications of extremal discs concern the study of moduli spaces (see for instance [16] and [3]) or the existence of a Green function for the homogeneous Monge-Amp`ere equation (see for instance [14, 20]).

Stationary discs are natural global biholomorphic invariants of complex manifolds with boundary. The smooth extension of stationary discs is an essential ingredient in boundary values problems such as the Fefferman extension theorem, as proved by L. Lempert [15] in the complex setting (see [6,8] for almost complex manifolds).

Lempert’s results were generalized to strongly pseudoconvex domains in the complex Euclidean space by E. Poletsky [21] and M. Y. Pang [19] using variational principles. Finally A. Tumanov [24] gave a geometric characterization of the sta- tionarity condition: a holomorphic disc f attached to the boundary of a domain is stationary if and only if f admits a meromorphic lift to the cotangent bundle with boundary attached to the conormal bundle over.

The second author’s research is supported by the Grant KRF-2008-357-C00006 of the Korea Research Foundation.

Received July 20, 2009; accepted in revised form November 13, 2009.

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Although there exist foliations by stationary discs in the almost complex setting [7,23], we first prove in this paper that in contrast with the complex setting governed by Lempert’s theory there exist extremal discs which are not stationary discs. Our main goal is to provide a new analytic and geometric characterization of extremal discs for smooth almost complex structures which are close to the standard complex structure on the closure of a given convex domain in the Euclidean space

R

2n. We deduce from this characterization the existence of an extremal disc through each point in any direction.

1.1. Organization of the paper

Our approach is based on a variational principle in the spirit of [19,21]. Throughout the paper we consider a bounded domain = {ρ < 0}contained in the almost complex manifold(

R

2n,J), whereJis a smooth almost complex structure on

R

2n. We assume that J is sufficiently close to the standard structure Jston some fixed neighbourhood of the closure offor an appropriate norm to be specified in the different sections.

In Section 2 we give an example of a proper extremal disc in the unbounded representation of the ball, endowed with a non-integrable almost complex structure, which is not stationary. Although the domain is not relatively compact the closed extremal disc is bounded and the domain is strongly pseudoconvex in a neighbor- hood of the closure of the disc.

The remaining part of the paper consists in finding a characterization of closed extremal discs, giving a substitute to the stationarity condition introduced by L. Lempert.

In Section 3 we study the properties ofJ-holomorphic variational vector fields.

These will be essential to characterize extremal discs.

In Section 4 we focus on extremal discs. We prove in Proposition 4.1 that every closed extremal disc contained inis proper. In Theorem 4.3 we prove that every closed extremal disc contained invanishes the first order variations. This technical result will give the necessary condition for a J-holomorphic disc to be extremal.

In Section 5 we find an analogue of the stationarity condition called J-sta- tionarity. We prove in Theorem 5.2 that a smooth closed extremal disc embedded intois J-stationary if the structureJ is close to the standard structureJst.

Finally in Section 6 we prove that every smooth J-stationary disc f is locally extremal if the domain is strongly convex in a neighborhood of f() (see Theo- rem 6.4).

ACKNOWLEDGEMENTS. We would like to thank Y. J. Choe for communicating his results concerning extremal discs in model almost complex manifolds.

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2.

A non stationary extremal disc

Throughout this paper, the summation convention will be used.

2.1. Preliminaries

We denote by

R

2n the 2n-dimensional real Euclidean space with standard coor- dinate system(x1, ...,x2n).The complex Euclidean space equipped with complex coordinates(z1, ...,zn)is denoted by

C

n. The real and complex coordinates satisfy the relation:

zj =x2j1+√

−1x2j

for j =1, ...,n. Analmost complex structureon an open setU of

R

2nis a smooth section J of the tangent bundleT U such that

Jp2= −I d|TpU

for every pU. We will mention the different precise smoothness assumptions on J in each result of the paper. The standard complex structure will be denoted by Jst, that is,

(

R

2n,Jst)=

C

n.

We denote by the open unit disc in

C

with standard complex coordinateζ = x +√

−1y. Let J be an almost complex structure on

R

2n. A smooth mapping f :

R

2n is called a J -holomorphic discif it satisfies the Cauchy-Riemann equation

∂f

∂x +J(f)∂f

∂y =0

on. Moreover if f is continuous (respectivelyCα-smooth for some realα >0) up to the boundary, then we call f a closed (respectivelyCα-closed) J-holomorphic disc.

A real-valued smooth functionρis said to beJ -plurisubharmonicif ddJcρ(X,J X)≥0

for every tangent vector X, wheredcJρ(X):= −dρ(J X). In case the inequality is strict whenever X =0, we say thatρisstrictly J -plurisubharmonic. It is known thatρ◦f is subharmonic onifρisJ-plurisubharmonic and f is aJ-holomorphic disc.

2.2. Extremal and stationary discs

Letρbe aC2-smooth function on

R

2n, let= {ρ <0}be a bounded domain in

R

2nand let J be aC1-smooth almost complex structure on

R

2n.

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Definition 2.1. A J-holomorphic disc f : is extremal if for every J- holomorphic discg :such thatg(0)= f(0)andg(0)=λf(0)for some λ≥0, it holds thatλ≤1.

To describe extremal discs, L. Lempert [14] introduced the notion of stationary discs on strongly convex domain in

C

n. LetT

R

2nbe the cotangent bundle of

R

2n. For a covectorω= pid xi, we consider the coordinatesω:(x1, ...,x2n,p1, ...,p2n) inT

R

2n. Thecanonical liftof J(in the terminology of [23], see [11] for a precise definition and properties of this lifted structure) is the almost complex structure on T

R

2ndefined by:

J:=

J 0 L Jt

where fori,j =1, ...,2n, Li j = −1

2pa

∂Jia

∂xj + ∂Jaj

∂xi +Jla ∂Jil

∂xmJmj∂Jlj

∂xmJim

.

Definition 2.2. A properly embedded closedJ-holomorphic disc f : is a stationary discif there exists a continuous mapu :

R

2n, of classC1on, and a continuous positive function ponsuch that

f˜:=(f,u) is J-holomorphic onand

u(ζ )=ζp(ζ )dρ(f(ζ)) (2.1) for everyζ. The map f˜is called theholomorphic liftof f.

Here the multiplication of a covectorω= pid xi byζ =x+√

−1yis defined byζω=(x pj +y Jjipi)d xj.

Remark 2.3.

1. A.Tumanov [24] generalized L. Lempert’s idea, considering meromorphic lifts of discs to the cotangent bundle, which are attached to the conormal bundle.

In definition 2.2 we define stationary discs in almost complex spaces in terms of holomorphic lifting. In [6] and [23], following [24], the authors used mero- morphic lifts instead of holomorphic lifts to define stationary discs. They call a closed J-holomorphic disc f which is attached to the boundary stationary if there exists a continuous lifted map (f,u˜) : \ {0} → T

R

2n which is J-holomorphic on\ {0}, such that 0 = ˜u(ζ ) is parallel with dρ(f(ζ )) for ζ and has singularity at the origin of order at most 1. This definition coincides with Definition 2.2. Indeed it follows from the definition of J˜ that if u := ζu, then˜ f˜ = (f,u)is still J -holomorphic. Applying the removable singularity theorem, f˜becomes a holomorphic lift of f.

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2. Spiro [22] proved that the conormal bundle

N

:= {tdρ(z) : t

R

\ {0},z

∂}is a totally real submanifold inT

R

2n with respect to J, if is strongly pseudoconvex with respect to J. Since(f,u˜)isJ˜-holomorphic on\{0}and is attached to the conormal bundle, one can see that(f,u˜)and hence f˜are smooth up toby a standard elliptic bootstrapping method. For more details on this subject, see [6, 23].

IfJ = Jst, then(f,u)isJ-holomorphic if and only if both f anduare holomorphic in the standard sense. IfJis a general almost complex structure this is no more true.

However we have:

Proposition 2.4. Let f :

R

2n be a J -holomorphic disc. Suppose(f,u) : T

R

2nis J -holomorphic. Then the complex pairing(u| f)defined by

(u| f):=ui∂fi

∂x −√

−1Jij(f)ui∂f j

∂x is a holomorphic function on.

Proof. For the sake of simplicity, we will use the notationhx andhy instead of

∂h/∂xand∂h/∂yfor a functionhon. Let

U :=Re(u| f)=uifxi and

V :=Im(u| f)= −Jij(f)ui fxj = −ui fyi. Since(f,u)is J-holomorphic,

ui x+Jijuj yukLki jfyj =0 (2.2) and

ui yJijuj x+ukLki jfxj =0 (2.3) for everyi =1, ...,2n, where

Lki j = 1 2

∂Jik

∂xj +∂Jkj

∂xi +Jlk ∂Jil

∂xmJjm∂Jlj

∂xmJim

.

Differentiating the Cauchy-Riemann equation for f, we get fx xi + Jij fyxj + ∂Jij

∂xk fyjfxk =0 (2.4) and

fyyiJij fx yj∂Jij

∂xk fxj fyk =0 (2.5) for everyi =1, ...,2n.

UxVy =(ui x fxi +ui yfyi)+ui(fx xi + fyyi ). (2.6)

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By (2.2) and (2.3),

ui x fxi +ui yfyi = −Jijuj yfxi +ukLki jfyj fxi +Jijuj xfyiukLki jfxjfyi

= −(ui x fxi +ui yfyi)+2uk fxi fyjLki j sinceLki j = −Lkj i. Therefore,

ui xfxi +ui yfyi =ukfxifyjLki j =ukfxifxlJlj Lki j. (2.7) From (2.4) and (2.5),

ui(fx xi + fyyi )= ∂Jji

∂xk∂Jki

∂xj

ui fxj fyk = ∂Jji

∂xk∂Jki

∂xj

ui fxj fxlJlk. (2.8) Plugging (2.6) with (2.7) and (2.8), we have

UxVy =ukfxifxlAkil where

Akil= Jlj

Lki j+ ∂Jik

∂xj∂Jkj

∂xi

.

Since Akil= −Akli from the definition ofLki j, it holds that UxVy =0.

Similarly we prove thatUy +Vx =0.

Corollary 2.5. Let= {ρ <0}be a bounded smooth domain in

R

2ndefined by a J -plurisubharmonic functionρ. Suppose f is a stationary disc ofwith holomor- phic lift f˜=(f,u). Then the complex pairing(u| f)is a positive constant.

Proof. Letζ =exp(

−1θ). Sinceρf is subharmonic, d

drρf(rexp(

−1θ))|r=1 = ζdρ(f(ζ )), f(ζ )>0

by the Hopf lemma, where·,· =Re(·|·)denotes the real pairing between a cov- ector and a vector. Sinceρf ≡0 on,

Imdρ(f(ζ))| f(ζ ))= − d

dθρf(ζ )=0.

Therefore,(u| f)is a holomorphic function onwhich is positive real-valued on

. This yields the conclusion.

Hence multiplying by a suitable positive constant, we may always assume that

(u| f)≡1, (2.9)

if f is a stationary disc and f˜=(f,u)is the holomorphic lift of f to the cotangent bundleT

R

2n.

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2.3. An example of an extremal disc which is not stationary

When J = Jst, it is known that everyC1-closed embedded extremal disc into a bounded strongly pseudoconvex domainis proper and stationary by Poletsky [21]

and Pang [19]. In the remaining part of this Section we provide a counterexample in the almost complex setting.

Fort >0, let Jt be an almost complex structure on

R

6defined by:

Jt :=







0 −1 0 0 0 t x3 1 0 0 0 t x3 0 0 0 0 −1 0 0

0 0 1 0 0 0

0 0 0 0 0 −1

0 0 0 0 1 0





 .

Let = {ρ(z) = Rez1+ |z2|2+ |z3|2 < 0} ⊂

R

6. Lee [13] showed thatis a homogeneous strongly Jt-pseudoconvex domain for everyt > 0 and that every strongly pseudoconvex domain with noncompact automorphism group in an almost complex manifold of real dimension 6 is biholomorphic to(,Jt)for somet. Proposition 2.6. The disc f(ζ) := (−1, ζ,0)is Jt-holomorphic and extremal in , but not stationary.

Proof. The disc f is obviously smooth up to the boundary. Choe [5] proved that there exists a uniquely determined Jt-Hermitian invariant metric which coincides with the standard Bergman metric at(−1,0,0)and has a constant negative holo- morphic sectional curvature−4. From the Ahlfors-Kobayashi Schwarz lemma, this implies that f is a Jt-holomorphic extremal disc infor everyt >0.

The canonical liftJt ofJt toT

R

6is given by the(12×12)-real matrix:

Jt =



















0 −1 0 0 0 t x3 0 0 0 0 0 0

1 0 0 0 t x3 0 0 0 0 0 0 0

0 0 0 −1 0 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0 0 0

0 0 0 0 0 −1 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 −1 0 0 0 0 0

0 0 0 0 −t p22t p21 0 0 0 1 0 0 0 0 0 0 t p21t p22 0 0 −1 0 0 0 0 0 t p22t p21 0 0 0 t x3 0 0 0 1 0 0 t p21 t p22 0 0t x3 0 0 0 −1 0

















 .

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Assume by contradiction that f is stationary. Let f˜=(f,u)be the holomor- phic lift of f toT

R

6. From the boundary condition (2.1) satisfied byuand from Condition (2.9), we have for(x,y):

u(x,y)=x 2,y

2,1,0,0,0

. (2.10)

Letu=(a,b,1,0,c,d)wherea,b,canddare real functions onby (2.9). Then

ay =bx, by = −ax, (2.11)

cy = tb

2 +t xbx +dx (2.12)

and

dy = ta

2 +t xaxcx (2.13)

by (2.2) and (2.3). Equalities (2.10) and (2.11) imply that a(x,y) = x/2 and b(x,y)= −y/2 on. Therefore, we have

cy =dxt y/4 anddy = −cx +3t x/4 (2.14) by (2.12) and (2.13). We deduce from (2.14) thatd is harmonic and henced ≡ 0 onby the boundary condition (2.10). Finally, we obtain from (2.14):

c=3t x2/8−t y2/8+C

for some real constantC. This contradicts the boundary condition (2.10).

Remark 2.7. The non-stationarity of the J-holomorphic extremal disc f(ζ ) = (−1, ζ,0)relies on the fact that the space ofJ-holomorphic variational vector fields along f is not stable under multiplication by J (see the next section for a precise definition of holomorphic variational vector fields). One can prove that if this space is stable under multiplication by J then the corresponding extremal disc is station- ary.

3.

Holomorphic variational vector fields

Letα > 2 be a non-integer real number. Let J be aCα-smooth almost complex structure on

R

2n. Let ft be a one parameter family of J-holomorphic discs into (

R

2n,J), of classC1int

R

. Consider forζ

v(ζ ):= d

dt ft(ζ )|t=0. From the J-holomorphicity of ft, we have

∂fti

∂x +Jij(ft)∂ftj

∂y =0 (3.1)

for everyi =1, ...,2n.

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Differentiating (3.1) int att =0, we get:

∂vi

∂x +Jij(f)∂vj

∂y +∂Jij

∂xk(f)∂f j

∂y vk =0 (3.2)

for everyi =1, ...,2n, where f = f0.

Definition 3.1. Let f be aJ-holomorphic disc in(

R

2n,J). AC1-smooth mapping v :

R

2n is called a J -holomorphic variational vector field along f if it satisfies equation (3.2).

We summarize basic properties of J-holomorphic variational vector fields.

Proposition 3.2. Let f : (

R

2n,J)be a J -holomorphic disc. Then the fol- lowing holds.

(a) If u andvare J -holomorphic variational vector fields along f , then so is u+v. (b) Let J1be an almost complex structure on

R

2mand let:U f()

R

2m be a pseudo-holomorphic map defined on an open neighborhood U in

R

2n, that is satisfying dJ = J1d. Then d(v)is a J1-holomorphic variational vector field alongf whenevervis a J -holomorphic variational vector field along f .

(c) The velocity vector field f := ∂f / ∂x of f is a J -holomorphic variational vector field along f .

(d) Moreover, ifφ = a+√

−1b:

C

is a holomorphic function, then so is φ f:=a f+b J f.

A proof of Proposition 3.2 is straightforward from the definition of J-holomorphic variational vector fields.

Remark 3.3.

1. The definition of J-holomorphic variational vector fields is related to the com- plete liftJofJto the tangent bundleT

R

2n. Let(xi,qi)be the naturally induced coordinates ofT

R

2n. Then

J := Jijd xj

∂xi +Jjidqj

∂qi +qk∂Jij

∂xkd xj

∂qi.

Thenvis a J-holomorphic variational vector field along a J-holomorphic disc f if and only if fˆ := (f, v)is a J-holomorphic disc into the tangent bundle.

For more details on complete lifts, one may refer to [11] and to [17].

2. Note that (d) is a special property of the velocity vector field f. In general, we cannot produce another J-holomorphic variational vector field by multiplying a given J-holomorphic variational vector fieldvby a holomorphic function.

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We have:

Proposition 3.4. Let f :

R

2n be a Cα-closed J -holomorphic disc. Let

V

be a small neighborhood of f(). There existsε >0such that ifJJstCα−1(V)< ε then for every J -holomorphic variational vector fieldvalong f of class Cα−1 on and satisfyingv(0)= 0there exists a one parameter family ht of Cα−1-closed

J -holomorphic discs of such that

h0= f ,

• d

dt|t=0ht =v,

ht(0)= f(0)+tv(0).

Proof. We may assume that f is an embedding. Otherwise, consider the(Jst×J)- holomorphic discF(ζ):=(ζ, f(ζ)):

C

×

R

2nand the(Jst×J)-holomorphic variational vector fieldV(ζ ):=(ζ, v(ζ))alongFsuch thatV(0)=0.

Letu2, ...,un beC smooth vector fields on a neighborhood of f() such that f,u2(f), ...,un(f)are

C

-linearly independent for everyζ. Consider the Cα-smooth diffeomorphism

:(ζ, w2, ..., wn)f(ζ )+ n

j=2

wjuj(f(ζ ))

defined on1(

V

), a neighborhood of f0(), where f0(ζ )=(ζ,0, ...,0). If we set J :=d1JdthenJJstCα−1(−1(V))is sufficiently small for an appro- priate choice ofu2, . . . ,un. Moreover J = Jstalong f0(). Finallyd1(f)(v) is a J-holomorphic variational vector field along f0ifvis aJ-holomorphic varia- tional vector field along f. We identify f with f0,J with J,vwithd1(f)(v) and

V

with1(

V

)via. Then J = Jstalong f0().

Let v be a J-holomorphic variational vector field along f0 with v(0) = 0, which isCα−1smooth up to. Thenvsatisfies the equation:

∂v

∂ζ¯ +Qv=0, (3.3)

where ∂v

∂ζ¯ = 1 2

∂v

∂x + Jst∂v

∂y

and

Qij(ζ):= 1 2

∂J2i

∂xj(f0(ζ )). (3.4) Let

Sv:=v+T0Qv (3.5)

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whereT0v(ζ )= 21πi

v(η)

η−ζv(η)η

¯is a solution operator for∂/∂ζ(see [25]). ThenSis an isomorphism of classCα−1on()which is close to the identity map, and Sv is a Jst-holomorphic map vanishing at 0 ∈ if and only if vis a

J-holomorphic variational vector field along f0vanishing at 0∈. Let

A

be the space of triples(J, v, φ)satisfying:

J is aCα−1-smooth almost complex structure on

V

such that J = Jstalong f0,

vis a J-holomorphic variational vector field along f0of classCα−1, bounded by 1, such thatv(0)=0,

φ:(−, )×

R

2nisC1-smooth,φ(t,·)is of classCα−1for eacht and φ(0,·)= f0,

φ˙ =v,φ(t,0)=0,

∂φ/∂x(t,0)=∂f0/∂x(0)+t∂v/∂x(0)for everyt for some small >0, whereφ˙ =dφ/dt|t=0.

LetG be the functional defined on

A

by:

G(J, v, φ)=Jφ= 1 2

∂φ

∂x +J(φ)∂φ

∂y

.

Then the target space ofGis:

B

:= {ψ:(−, )×→

R

2n:ψof classC1withψ(t,·)of classCα−2for eacht, ψ(0,·)=0, ψ˙ =0}.

Letφ0= f0. Then

G(Jst,0, φ0)=0. The differentialL:=∂G/∂φat(Jst,0, φ0)is

L=Jst

from

C

:= {ξ:(−, )×

R

2n : ξof classC1withξ(t,·)of classCα−1for eacht, ξ(0,·)= ˙ξ ≡0 and

ξ(t,0)=∂ξ/∂x(t,0)=0 for everyt}.

to

B

. LetR:

B

C

be a bounded operator defined by Rψ(t, ζ )=Tψ(t, ζ)Tψ(t,0)

∂ζTψ(t,0)

ζ

for everyψ

B

where

Tψ(t, ζ)= 1 2πi

ψ(t, η)

ηζ dη∧dη.¯

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Then R is a bounded right inverse of L. It follows from the Implicit Function Theorem that for every J sufficiently close to Jst inCα−1-norm and for every J- holomorphic variational vector field vsufficiently small in Cα−1-norm there is a one parameter familyφt of classCα−1onsatisfying the conclusions of Proposi- tion 3.4. Finally for a given J-holomorphic variational vector fieldvchoosec>0 such thatcvis sufficiently small inCα−1norm. Consider the one parameter family φt associated tocvand setht :=φt/c.

4.

First order variations

In this section we give a necessary condition for a J-holomorphic disc to be ex- tremal in terms of J-holomorphic variational vector fields. Letα be a non-integer real number larger than 2. We assume thatJ is aCα-smooth almost complex struc- ture on

R

2n.

Proposition 4.1. Letbe a bounded domain in

R

2n defined by a C2-smooth J - plurisubharmonic function ρ. Let f : be a Cα-closed extremal disc.

SupposeJJstCα−1 < εwhereεis given by Proposition 3.4. Then f is attached to the boundary of, that is f(∂)∂.

Proof. Suppose by contradiction that f is not attached to the boundary of. Then there exists an open intervalP inwith positive length and a compact subset K ofsuch that f(rζ)K for everyζPand for everyr ≤1, sufficiently close to 1. Choose a relatively compact sub-interval P1 in P and let 0 ≤ χ ≤ 1 be a fixed smooth function on , compactly supported in P and equal to one on P1. For a positive constant R, we denote byφR the holomorphic function onsuch that ReφR is the Poisson integral of and ImφR(0) = 0. For everyr < 1 let fr(ζ) := f(rζ )and letvrR(ζ) := ζ expR(ζ )) fr(ζ ).Here and throughout the paper we set f :=∂f / ∂xfor every map f defined on an open set of

R

2. ThenvrR

is aJ-holomorphic variational vector field along fr. It follows from Proposition 3.4 that there exists a one-parameter familyhrR,t of J-holomorphic curves defined for 0≤t <t0=t0(R)such that:

hrR,0= fr, d

dt hrR,t|t=0=vrR and

hrR,t(0)= fr(0)+tvrR(0)=r(1+t expR(0)))f(0)

for every R, r andt. We point out thatt0 depends a priori onr. So we to take the infimum overr < 1 close to 1. Since expR(0)) ≥ exp(l R/2π)wherel =

length of P1>0, ifr(1+texp(l R/2π)) >1, or equivalently if t >t1(r,R):= 1−r

r exp(−l R/2π),

thenhrR,t yields a contradiction to the extremality of f wheneverhrR,t().

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Note that for every J-holomorphic curve g : the functionρg is subharmonic. So there exists a constantC1>0 depending only onρandg(0)such that

ρg(ζ )≤ −C1(1− |ζ|) for everyζ.

Letζ\P. Then|expR(ζ ))| = 1. Therefore, |d(ρhrR,t)/dt|t=0L()fL(). Since the right hand side of this inequality does not depend onR, there is a constantC2>0 independent ofRandt2(R) >0 such that

|ρ◦hrR,t(ζ)ρfr(ζ)| ≤C2t

for everyζ\Pand every 0≤t <t2(R).Therefore, ifC2tC1(1−r)or, t <t3(r):=C3(1−r),

thenhrR,t(ζ)for everyζ\P, whereC3 =C1/C2. We may assume that t3(r)t2(R)sincet3(r) → 0 asr → 1. Now suppose thatζP. Then there exist positive constantsd=d(K, ρ),C4(R)andt4(R)such that

ρfr(ζ)≤ −d and

|ρ◦hrR,t(ζ )ρfr(ζ)|<C4(R)t

for every t < t4(R). Therefore, if t < min{t4(R),t5(R) = d/C4(R)}, then hrR,t(ζ)for everyζP.

Choose Rsuch that

exp(−l R/2π) <C3/2. Then

t1(r,R) <t3(r)

for every 1/2<r <1. Finally, chooser close to 1 so that t3(r)≤min{t0(R),t2(R),t4(R),t5(R)}.

Then for everyt <t3(r),hrR,t mapsintoand if we chooset1(r,R) <t <t3(r), thenhrR,t yields a contradiction for the extremality of f.

Definition 4.2. A Cα-closed J-holomorphic disc f : attached to the boundary ofis said to vanish the first order variations if for everyJ-holomorphic variational vector fieldualong f, of classCα−1on, such that:

(a) dρ(f(ζ)),u(ζ) =0 for everyζ, (b) u(0)=0 and

(c) ∂u/∂x(0)=λ ∂f/∂x(0)for aλ

R

, it holds thatλ=0.

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We have the following necessary condition for a J-holomorphic disc to be extremal:

Theorem 4.3. Let be a bounded domain in

R

2n defined by a C2-smooth J - plurisubharmonic functionρ. Let f be a Cα-closed extremal disc in. IfJJstCα−1(f()) < εfor a sufficiently smallε > 0, then f vanishes the first order variations.

Proof. We already proved in Proposition 4.1 that f is attached to the boundary of . Suppose that f does not vanish the first order variations. It follows from Definition 4.2 that we may choose aJ-holomorphic variational vector fieldualong

f such that:

• dρ(f(ζ)),u(ζ ) =0 for everyζ,

u(0)=0,

u(0)=λf(0)for someλ

R

\ {0}.

We may assume thatλ >0 by taking−u, if necessary. Let v(ζ ):=u(ζ )λ

2ζ f(ζ ).

Thenvis also aJ-holomorphic variational vector field along f of classCα−1on by Proposition 3.2. Choose a 1-parameter family ft forvsatisfying the properties of Proposition 3.4. Then for everyζ:

d

dtρft(ζ)|t=0 = dρ(f(ζ )),u(ζ) − λ 2

dρ(f(ζ )), ζ ∂f

∂x(ζ )

= dρ(f(ζ )),u(ζ ) − λ 2

d

drρf(rζ )|r=1

< 0

by the hypothesis and by the Hopf lemma. Sinceρf vanishes identically on, thenρft <0 onift >0 is small enough. hence ft is a J-holomorphic disc intofor every sufficiently smallt >0. On the other hand,

ft(0) = f(0)+tv(0)

=

1+λt 2

f(0).

This yields a contradiction to the extremality of f by choosing smallt >0.

5. J

-stationary discs

Letαbe a non integer real number larger than 2. Let = {ρ <0}be a bounded domain in

R

2n where ρ is a Cα+1-smooth J-plurisubharmonic function. Let J be a Cα-smooth almost complex structure on

R

2n and let f be a Cα-closed J- holomorphic disc in.

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5.1. Extremal discs are J-stationary

Suppose f is embedded and extremal in. Then f vanishes the first order vari- ations by Theorem 4.3. If vis a J-holomorphic variational vector field along f, Cα-smooth on, such thatv(0)=0,v(0)=λf(0)for some realλand

dρ(f(ζ)), v(ζ) =0 (5.1)

for everyζ, then it holds thatλ=0.

Let pbe a positive function defined for everyζby:

1 p(ζ) = d

drρf(rζ )|r=1= ζdρ(f(ζ )), f(ζ ).

Letvbe a J-holomorphic variational vector field along f,Cα-smooth on, and such that v(0) = v(0) = 0. There is a unique holomorphic function h on , Cα-smooth on, and such that

Reh= −pdρ(f), v1 onand

Imh(0)=0.

Thenv1(ζ):=v(ζ)+ζh(ζ )f(ζ)defines aJ-holomorphic variational vector field along f,Cα-smooth on, which satisfies equation (5.1) and such thatv(0) = 0, v(0)= h(0)f(0). Hence, we see that f vanishes the first order variations if and only if

0= −λ= −h(0)= −Reh(0)=

p(eiθ)dρ(f(eiθ)), v(eiθ)dθ

2π (5.2) for every J -holomorphic variational vector fieldvwhich is smooth on, such that v(0)=v(0)=0.

Now suppose thatJ is sufficiently close to JstinCαnorm, in a neighborhood of f(). Then every J-holomorphic vector field along f which isCα-smooth on is completely determined by its boundary values by the classical Schauder theory of linear elliptic partial differential equations. Hence, we may regard the space of J-holomorphic vector fields along f which are Cα-smooth on as a subspace ofCα(∂). We denote by

A

2the L2(∂)-closure of the space of J-holomorphic variational vector fieldvalong f which areCα-smooth up to the boundary and such thatv(0)=v(0)=0. Let

H

2:=

k2

akei kθL2(∂):ak

C

n

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be a subspace of the classical Hardy space. Recall thatvis a J-holomorphic varia- tional vector field along f if it satisfies:

∂v

∂x +J(f)∂v

∂y +Wv=0 whereW =(Wij)is given by

Wij = ∂Jki

∂xj(f)∂fk

∂y for everyi,j =1, ...,2n. Let

S˜v= ˜SJ,fv:=v+KvKv(0)

∂ζKv(0)

ζ

forvCα(), whereK =KJ,f is a linear operator defined by Kv:= 1

2T

(J(f)Jst)

∂y +W

v

andT is the Cauchy-Green operator defined by T u(ζ )= 1

2πi

u(η)

ηζdηdη.¯

Fix aCα-closed Jst-holomorphic disc f0. ThenS˜ is a bounded operator onCα() to itself, which tends to the identity operator as f converges to f0inCαnorm on and J converges to JstinCαnorm on a neighborhood of f0(). Moreover,vis a J-holomorphic variational vector field along f satisfyingv(0)=0 andv(0)= 0 if and only if S˜v is Jst-holomorphic and satisfies S˜v(0) = 0 and (S˜v)(0) = 0.

Therefore,

A

2Cα()is isomorphic to

H

2Cα()and they are close to each other inCα-sense. This implies that

A

2is isomorphic to

H

2, and is a small perturbation of

H

2inL2(∂). Moreover sinceT is a bounded operator fromCα−1()toCα() then for a fixedv, the correspondence(J, f)→ ˜SJ,fvisC1-smooth. This implies that

A

2Cα()variesC1-smoothly as(J, f)changes inCα-space.

Since

A

2 is sufficiently close to

H

2 in L2(∂), the L2-projection π : L2(∂)

H

2 induces an isomorphism from

A

2 onto

H

2. Let

H

2 be the or- thogonal complement of

H

2in L2(∂)and letbe the isomorphism of L2(∂) onto itself defined by

(v):=

v ifv

H

2

π(v) ifv

A

2. (5.3)

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If f vanishes the first order variations, then from equation (5.2) we have:

0 =

pdρ(f), vdθ

=

(1)(pdρ(f)), (v)dθ

for everyv

A

2, where(1)is theL2-adjoint of1. By the definition of, (v)/ζ2 is an arbitrary standard holomorphic map contained in L2(∂). There- fore, f vanishes the first order variations if

ζ(1)(pdρ(f)),hdζ =0

for every standard holomorphic map hL2(∂). This implies that the cor- respondence ζζ(1)(pdρ(f))(ζ ) extends continuously to a Jstt- holomorphic mapping defined on . Following a viewpoint of [14], we define

J-stationary discs for a structureJ sufficiently close toJstas follows:

Definition 5.1. Letα > 1, non integer. Let = {ρ < 0}be a bounded domain with ρof class Cα+1. Let f be a properly embedded Cα-closed J-holomorphic disc. Suppose J is sufficiently close to Jst in a neighborhood of f() in Cα norm so that the isomorphism in (5.3) is well-defined. Then we say that f is J -stationaryif there exists a positiveCα-smooth map pdefined onsuch that the correspondence ζζ(1)(p dρ(f))(ζ )continuously extends to a Jstt-holomorphic mapping defined on. Here, the multiplication byζ to 1-forms is defined by Jstt.

It should be mentioned that the positive function p in Definition 5.1 may be modified by multiplying a given one by real positive constants. Hence we may assume that p(1)=1.

We have proved the following theorem:

Theorem 5.2. Let J be a C-smooth almost complex structure on

R

2n and let be a bounded domain with a C-smooth J -plurisubharmonic defining functionρ. Let f be a C-closed extremal disc embedded into. Suppose that J is sufficiently close to Jstin a neighborhood of f()in Cα norm (α >1, non-integer). Then f is J -stationary.

Proof. According to Theorem 4.3 f vanishes the first order variations. It follows from the discussion preceding Theorem 5.2 that f isJ-stationary.

5.2. Existence of J-stationary discs

If J = JstandDis a strongly convex domain in

C

n, it is known by [14] that there exists a unique Jst-stationary disc f0 such that f0(0) = 0 and f(0) = λ0V for someλ0>0. We prove the following:

Références

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