L’INSTITUTFOURIER DE ANNALES

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A L

E S

E D L ’IN IT ST T U

F O U R

ANNALES

DE

L’INSTITUT FOURIER

Sergey IVASHKOVICH & Alexandre SUKHOV

Schwarz Reflection Principle, Boundary Regularity and Compactness for J-Complex Curves

Tome 60, n^o4 (2010), p. 1489-1513.

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SCHWARZ REFLECTION PRINCIPLE, BOUNDARY REGULARITY AND COMPACTNESS FOR J-COMPLEX

CURVES

by Sergey IVASHKOVICH & Alexandre SUKHOV

Abstract. — We establish the Schwarz Reflection Principle for J-complex discs attached to a real analyticJ-totally real submanifold of an almost complex manifold with real analyticJ. We also prove the precise boundary regularity and derive the precise convergence in Gromov compactness theorem inC^k,α-classes.

Résumé. — Nous établissons le Principe de Symétrie de Schwarz pour les disques complexes attachés à une sous-variété analytique réelle et totalement réelle d’une variété presque complexe munie d’une structure presque complexe analytique réelle. Nous prouvons également la régularité au bord précise de ces disques et nous en déduisons la convergence exacte dans le théorème de compacité de Gromov dans les classesC^k,α.

1. Introduction 1.1. Reflection Principle

Denote by ∆ the unit disc inC, by S- the unit circle. Let β ⊂Sbe a non-empty open subarc ofS.

Theorem 1.1 (Reflection Principle). — Let (X, J) be a real analytic almost complex manifold andW a real analyticJ-totally real submanifold ofX. Letu: ∆→X be aJ-holomorphic map continuous up toβ and such thatu(β)⊂W. Thenuextends to a neighborhood ofβas a (real analytic) J-holomorphic map.

Keywords:Almost complex structure, totally real manifold, holomorphic disc, reflection principle.

Math. classification:32Q65, 32H40.

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The case of integrableJ is due to H. A. Schwarz [15]. Indeed, one can find local holomorphic coordinates in a neighborhood ofu(p)for a takenp∈β such that W =Rⁿ in these coordinates and now the Schwarz Reflection Principle applies. In our case there is no such reflection, since a general almost complex structure doesn’t admits any local (anti)-holomorphic maps.

But the extension result still holds.

One can put Theorem 1.1 into a more general form of Carathéodory, [3].

For this recall that the cluster setcl(u, β) of u at β consists of all limits limk→∞u(ζk) for all sequences {ζk} ⊂ ∆ converging to β. In [5] it was proved that if the cluster setcl(u, β)of a J-holomorphic mapu: ∆→X is compactly contained in a totally real submanifoldW then u smoothly extends toβ. Therefore we derive the following

Corollary 1.2. — In the conditions of the Theorem 1.1 the assump- tion of continuity of u up to β and u(β) ⊂ W one can replace by the assumption thatu(∆)is compact and the cluster set cl(u, β)is contained inW.

1.2. Boundary Regularity

For the proof of our Reflection Principle we need to study not only real analytic boundary values but also the smooth ones (with finite smoothness). For our method to work we need the precise regularity and a certain kind of uniqueness of smoothJ-complex discs attached to a J-totally real submanifold. The result obtained is the following

Theorem 1.3 (Boundary Regularity). — Letu: (∆, β)→(X, W)be aJ-holomorphic map of classL^1,2∩ C⁰(∆∪β), whereW isJ-totally real.

Then:

(i) for any integerk>0and real0< α <1ifJ ∈ C^k,αandW ∈ C^k+1,α thenuis of classC^k+1,α on∆∪β;

(ii) fork>1 the conditionu∈L^1,2∩ C⁰(∆∪β)andu(β)⊂W can be replaced by the assumption thatu(∆) is compact and the cluster setcl(u, β)is contained inW.

Remark 1.4. — IfJ is of classC⁰ andW ofC¹ thenu∈ C^αup toβ for all0< α <1. This was proved in [10], Lemma 3.1.

For integrable J the result of Theorem 1.3 is due to E. Chirka [4]. For non-integrableJ weaker versions of this Theorem were obtained in [5, 7, 12].

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Namely, theC^k,α-regularity ofuup toβ was achieved there under the same assumptions. The precise inner regularity was obtained by J.-C. Sikorav in [16].

1.3. Compactness Theorem

The precise regularity of J-complex discs attached to a J-totally real submanifolds of Theorem 1.3 allows also to get the precise convergence in Gromov compactness theorem. We refer to the Subsection 6.1 and to [9, 10]

for the relevant notions and definitions.

Theorem 1.5 (Compactness Theorem). — Let a sequence {Jn} of al- most complex structures of classC^k,α,k>0,0< α <1, on a Riemannian manifold (X, h) converge on a compact subset K ⊂ X in C^k,α-topology to an almost complex structureJ. Let a sequence Wn ={(Wi, fn,i)}^m_i=1 ofJ_n-totally real immersed submanifolds of X of classC^k+1,α converge in C^k+1,α-topology to a J-totally real immersion W = {(Wi, fi)}^m_i=1. Sup- pose that allW_n andW have only weak transverse self-intersections. Let furthermore, {( ¯Cn, un)} be a sequence of stable Jn-complex curves over X, parametrized by a fixed oriented compact real surface with boundary Σ = (Σ, ∂Σ), such that:¯

(i) un(Cn) ⊂ K and that there exists a constant M such that area[un(Cn)]6Mfor alln;

(ii) ( ¯C_n, u_n)satisfy the totally real boundary conditions(W_n,β,u^(b)n ) withu^(b)_n,i(β_i)⊂W_i for allnandi.

Then there exits a subsequence {(Cn_k, un_k)} of {(Cn, un)} and para- metrizationsσn_k : Σ→Cn_k, such that(Cn_k, un_k, σn_k)converges inC^k+1,α- topology up to boundary to a stableJ-complex curve(C, u, σ)overX and this(C, u)satisfies the totally real boundary conditions(W,β,u^(b))with someC^k+1,α-continuous mapsu^b_k:β_k →W_i.

The novelty here with respect [12, 6] is that there is no loss in both of regularity and convergence of curves up to the boundary. In [10] an analog of Theorem 1.5 was proved for the special casek=α= 0.

1.4. Proofs

The interior analyticity of J-holomorphic discs in analytic almost complex manifolds follows from classical results on elliptic regularity in the

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real analytic category, see, for instance, [2]. However the real analyticity up to the boundary is not a consequence of the known results since we do not deal with a boundary problem of the Dirichlet type. The direct application of the reflection principle (in the form of Vekua, for exam- ple) also leads to technical complications because of the non-linearity of the Cauchy-Riemann operator on an almost complex manifold. So our ap- proach is different and is based on the reduction of the boundary regularity to a non-linear Riemann-Hilbert type boundary-value problem.

This paper is organized in the following way.

1. In §3, using [10] and [7] we prove Theorem 1.3. First we establish theC^1,α- regularity ofuif J ∈ C^α and then, using a sort of “geometric bootstrap”, we obtain theC^k+1,α-regularity ofuifJ ∈ C^k,α.

2. In §4 we prove the solvability and uniqueness of a Riemann-Hilbert type boundary-value problem in Sobolev classes - the principal new tool of this paper. In §5 we adapt this method to the real analytic case. Then the uniqueness, both in Sobolev and in real analytic categories gives the proof of the Reflection Principle of Theorem 1.1. The novelty in §4 and 5, is a non-standard choice of the smoothness classes and an application of the Riemann-Hilbert boundary-value problem in the resolving of boundary regularity questions.

3. In §6 we recall necessary notions and prove the Compactness Theorem.

4. We end up with the formulation of open questions in §7.

We would like to express our gratitude to J.-F. Barraud who turned our attention to the question of extendability ofJ-holomorphic maps through totally real submanifolds in real analytic category.

2. Preliminaries

In what follows(X, J)will denote a pair which consists of a real analytic manifoldX and an almost complex structureJ on it. Note that by the well- known theorem of Whitney any smooth manifold can be endowed with a compatible atlas with real analytic transition mappings, i.e., the condition of real analyticity ofX doesn’t lead to any loss of generality. The regularity ofJ will be specified in each statement. ByJ_st =iIddenote the standard complex structure ofCⁿ(as well as ofCand of∆). Everywhere throughout the paper we use the notation L^k,p for Sobolev space of functions with generalized partial derivatives of classL^p up to the orderk. AC¹-map (or

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C⁰∩L^1,1_loc-map)u: ∆→X is called J-holomorphic if for every (or almost every)ζ∈∆

du(ζ)◦J_st=J(u(ζ))◦du(ζ) (2.1)

as mappings of tangent spaces Tζ∆ → T_u(ζ)X. The image u(∆)is called then aJ-complex disc. Every almost complex manifold (X, J)of complex dimensionncan be locally viewed as the unit ballBin Cⁿ equipped with an almost complex structure which is a small deformation of Jst. To see this fix a pointp∈X, choose a coordinate system such that p= 0, make anR-linear change of coordinates in order to have J(0) =Jst and rescale, i. e., consider J(tz) for t > 0 small enough. Then the equation (2.1) of J-holomorphicity of a map u: ∆→B can be written in local coordinates ζon∆andzonCⁿas the following first order quasilinear system of partial differential equations

u_ζ−AJ(u)u_ζ = 0, (2.2)

whereAJ(z)is the complexn×nmatrix of the operator whose composite with complex conjugation is equal to the endomorphism(J_st+J(z))⁻¹(J_st− J(z))(which is an anti-linear operator with respect to the standard struc- tureJst). SinceJ(0) =Jst, we have AJ(0) = 0. So in a sufficiently small neighborhood of the origin the normkAJ kL^∞ is also small which implies the ellipticity of the system (2.2).

We recall some classical integral transformations. Let Ωbe a relatively compact domain inCbounded by a finite number of transversely intersect- ing smooth curves. Denote byT_Ω^CGthe Cauchy-Green transform inΩ:

(2.3) T_Ω^CGh

(ζ) = 1 2πi

Z Z

Ω

h(τ)dτ ∧dτ τ−ζ . Denote also by

(2.4) K_Ωh

(ζ) = 1 2πi

Z

∂Ω

h(τ)dτ τ−ζ the Cauchy transform.

Proposition 2.1. — For every integerk>0, real0< α <1 and real p >1the following holds:

(i) T_Ω^CG:C^k,α( ¯Ω)−→ C^k+1,α( ¯Ω)(resp.T_Ω^CG:L^k,p(Ω)−→L^k+1,p(Ω)) is a bounded linear operator and(T_Ω^CGh)_ζ =hfor anyh∈ C^k,α(Ω) (resp. for anyh∈L^k,p(Ω)).

(ii) Lethbe a bounded real analytic function onΩ, thenT_Ω^CGhis real analytic onΩ. If, furthermore, Ω = ∆, his a sum of a converging

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seriesh(ζ, ζ) = Σh_k,lζ^kζ¯^land is continuous up to the boundary∂∆, then for anyζ∈∆we have

T_∆^CGh(ζ) =H(ζ, ζ)− K∆H (ζ), (2.5)

whereH is a primitive of hwith respect toζ.

The proof of (i) is contained, for instance, in [18], Theorems 1.32 and 1.37.

For the statement (ii) about real analyticity see [18], p. 26. Relation (2.5) is in fact nothing but the Cauchy-Green formula. We shall also need the Schwarz integral transform on∆:

(2.6) T^SWh

(ζ) = 1 2πi

Z

∂∆

τ+ζ τ−ζ ·h(τ)

τ dτ.

Denote by O^k,α(Ω) (resp. O^k,p(Ω)) the Banach space of holomorphic mapsg: Ω−→Cⁿ of classC^k,α( ¯Ω)(resp.L^k,p( ¯Ω)). This space is equipped with the normkgk_Ck,α( ¯Ω) (resp.kgk_Lk,p( ¯Ω)).

Proposition 2.2. — T^SW:C^k,α(S)−→ O^k,α(∆)(resp.T^SW:L^k,p(S)−→

O^k,p(∆)) and KΩ : C^k,α(∂Ω) −→ O^k,α(Ω) (resp. KΩ : L^k,p(∂Ω) −→

O^k,p(Ω)) are bounded linear operators. For any real-valued function ψ ∈ C^k,α(S)one has

(2.7) Re(T^SWψ)|S=ψ

and

Im(T^SWψ)(0) =0.

For the proof see, for instance, [18], Theorem 1.10.

We shall need to consider the space of traces of functions from L^1,p(∆) on the unit circle S, see [13]. We say that a function φ defined on S is in the space T^1,p(S), p > 2, if there exists a function u ∈ L^1,p(∆) such that u|S = φ. Let us point out that by the Sobolev imbedding we have T^1,p(S) ⊂ C^α(S) with α= (p−2)/p. On the other hand if 1/2 < α <1 then C^α(S) ⊂T^1,p(S) for every p < (1−α)⁻¹. Indeed, given φ∈ C^α(S) its Schwarz integralu=T^SWφ is of classO^α(∆) and so by the classical theorem of Hardy-Littlewood, see [13], one has

du(ζ) dζ

6 C (1− |ζ|)^1−α.

This implies that ^du(ζ)_dζ ∈L^p(∆)forp(1−α)<1. Now the Poisson integral (2.8) P φ:=Re T^SW(Reφ) +iRe T^SW(Imφ)

gives an extension ofφof classL^1,p(∆).

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Proposition 2.3. — Ifφ∈T^1,p(S)thenT^SWφ∈L^1,p(∆).

Proof. — Let u∈L^1,p(∆) be an extension ofφ. By the Cauchy-Green formula

(2.9) u=K_∆φ+T_∆^CGu_ζ.

Hence the Cauchy type integral in the right hand (and therefore the Schwarz integral) is of classL^1,p(∆), which proves the proposition.

Finally, we introduce the norm on the space T^1,p(S)by setting kφk_T1,p :=kP φk_L1,p(∆)

whereP φdenotes the Poisson integral (2.8). Obviously,T^1,p(S)is a Banach space. It follows from (2.8) that the convergence inC^α(S),α >1/2implies the convergence in T^1,p(S), i.e., C^α(S) is a closed subspace in T^1,p(S).

Furthermore, if a sequence of functions{un} converges inL^1,p, then their tracesφ_n:=u_n|Sconverge inT^1,p(S). This follows from the Cauchy-Green representation (2.9) since the convergence of the Cauchy type integral implies the convergence of the Schwarz and Poisson integrals.

Remark 2.4. — In conclusion of this section we point out that the no- tion of the trace space T^1,p(∂Ω)on the boundary of a domain Ω can be extended to a much larger class of simply connected domains by means of the Riemann mapping theorem and the classical theory of boundary properties of conformal mappings. For instance, all above definitions and properties admits an immediate generalization to the case where ∆ is replaced by a simply connected domain bounded by a finite number of real analytic arcs with transversal intersections. The special case of the upper semi-disc∆⁺ will be important for our considerations. In what follows we simply writeT^1,p in the case of the unit circle.

3. Boundary Regularity in Hölder Classes

In this section we shall first use a version of a Reflection Principle pro- posed in [10] to prove the casek= 0of Theorem 1.3. Then using the “geometric bootstrap” from [7] we obtainC^k+1,α-regularity of complex discs in C^k,α-regular structures for allk>1, thus proving the Theorem 1.3.

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3.1. Reflection Principle-I

In this subsection the structureJ is supposed to be of class C^αonly. A J-totally real submanifoldW ofX will be supposed to haveC^1,α-regularity.

Similarly to the integrable case, a real submanifoldW of an almost complex manifold(X, J) is called J-totally real if TpW ∩J(TpW) = {0} at every point pof W. If n is the complex dimension of X, then any totally real submanifold ofX is locally contained in a totally real submanifold of real dimensionn. So in what follows we assume thatW isn-dimensional.

First we make a suitable change of coordinates.

Lemma 3.1. — One can find coordinates in a neighborhood V of p ∈ W such that in these coordinates V = R²ⁿ, W = Rⁿ, J|_Rⁿ = J_st and J(x, y)−Jst =O(||y||^α).

Proof. — After a change of coordinates of classC^1,α we can suppose that in some neighborhood ofp= 0our manifoldW coincides withRⁿ. Next we are looking for a C^1,α-diffeomorphism ϕ= (ϕ1, ...ϕ2n) in a neighborhood of the origin such that

(1) ϕj(x,0) =xj forj = 1, ..., n;

(2) ϕ_j(x,0) = 0forj=n+ 1, ...,2n;

(3) _∂y^∂ϕ

j(x,0) =J(x,0)

∂

∂xj

forj= 1, ..., n.

SuchC^1,α-diffeomorphism exists due to the Trace theorem, see [17]. In new coordinates given by ϕ we shall clearly have W = Rⁿ, J|_Rn = J_st and J(x, y)−Jst =O(||y||^α)due toC^α-regularity ofJ.

In what follows the disc ∆ with an arc β on its boundary S will be suitable for us to change by the upper half-disc∆⁺={ζ:Reζ >0} and the segment(−1,1). Letu: (∆⁺, β)→(X, W)be aJ-holomorphic map of classL^1,pup toβ= (−1,1)for somep >2.

The following lemma will prove the casek= 0of Theorem 1.3 and will be used in the proof of the same case of Theorem 1.5 in the last section.

Lemma 3.2. — Let J be of class C^α and W be J-totally real of class C^1,α. Let u: (∆⁺, β)→(X, W)be J-holomorphic of class C⁰∩L^1,2 up to β. Thenuis of class C^1,α up toβ.

Proof. — We can assume thatW =Rⁿ andJ(x, y)−J_st=O(||y||^α). On the trivial bundle∆⁺×R²ⁿ→∆⁺we consider the following linear complex structure:J_u(z)[ξ] =J(u(z))[ξ]forξ∈R²ⁿ andz∈∆⁺. At this point we stress that Ju is defined only on ∆⁺ ×R²ⁿ. Denote by τ the standard

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conjugation in ∆ ⊂ Cas well as the standard conjugation in R²ⁿ = Cⁿ. Now we extendJu to∆×R²ⁿ by setting

(3.1) J˜u(z)[ξ] =−τ J_{u(τ z)}[τ ξ]forz∈∆andξ∈R²ⁿ.

We consider nowuas a section (over∆⁺) of the trivial bundleE=R²ⁿ×

∆→ ∆ and endowE with the complex structureJ˜u. Complex structure J˜_u defines a ∂-operator ∂J˜uw = ∂_xw+ ˜J_u∂_yw on L^1,p-sections of E for all 1 6 p < ∞ (for this only continuity of J˜u is needed). Remark that u is J˜u-holomorphic on ∆⁺. By F we denote the totally real subbundle

∆×Rⁿ→∆ ofE.

Definition 3.3. — Define the “extension by reflection” operatorext : L¹(∆⁺,E)→L¹(∆,E)by setting

(3.2) ext(w)(z) =τw(τz)

forz∈∆⁻ andw∈L¹(∆⁺, E). We shall also writew˜ forext(w).

Note that if w is continuous up toβ and takes on β values in the sub- bundleF thenext(w)stays continuous. By the reflection principle of The- orem 1.1 from [10] we know that ext : L^1,p(∆⁺,E,F) → L^1,p(∆,E) is a continuous operator for all16p <∞and that∂J˜_uw˜= 0if∂J_uw= 0. Let

˜

ube the extension of u, in particular, u˜ is J˜u-holomorphic on ∆ of class L^1,2(∆). First a priori estimate (1.1) from [9] insures thatu˜∈L^1,p_loc for all p <∞. In particular u˜ ∈ C^γ for everyγ < 1. Therefore J˜u is of class C^δ, whereδ=αγ.

Set v =ρ˜u, where ρis a cut-off function equal to1 in ∆1

2. Then from (2.2) we get

(3.3) vζ¯−AJ(˜u)v_ζ =g, where the functiong =

ρζ¯−ρζAJ(˜u)

˜

u is of class C^δ(∆). Observe that vζ¯−AJ(˜u)v_ζ =

v−T_∆^CGAJ(˜u)v_ζ

ζ. Elliptic regularity implies that v− T_∆^CGAJ(˜u)v_ζ is of classC^1,δand invertibility of the operatorId−T^CG_∆ AJ(˜u)_∂z^∂ inC^1,δ gives thatv is inC^1,δ.

One can repeat this step once more to getC^1,α-regularity ofv on∆ and

therefore ofuup toβ.

3.2. Cluster Sets on Totally Real Submanifolds

Since the case k = 0 of the Theorem 1.3 is already proved, we restrict ourselves in the future withk>1. Fix an almost complex manifold(X, J)

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withJ of classC^1,α and aJ-totally real submanifoldW of class C^2,α. Let u : ∆ → X be a bounded J-holomorphic map of the unit disc into X. Suppose thatcl(u, β)bW, whereβ is some non-empty open subarc of the boundary.

We use the Proposition 4.1 from [5] and observe that u is in Sobolev classL^1,p up toβ for allp <4. In particularuisC^β-regular up toβ with β= 1−_p² (this means for allβ < ¹₂). Lemma 3.2 implies now the following Corollary 3.4. — Let J ∈ C^1,α and W ∈ C^2,α. If u : (∆⁺, β) → (X, W) is a bounded J-holomorphic map with cl(u, β) b W then u ∈ C^1,α(∆⁺∪β).

Let’s stress here thatC^1,αis not the optimal regularity ofu, it should be C^2,α. This will be achieved in the next subsection.

3.3. Geometric bootstrap and boundaryC^k+1,α-Regularity Having proved the C^1,α-regularity of complex discs in C^α-regular structures we are going to use the “geometric bootstrap” to obtain C^k+1,α- regularity of complex discs inC^k,α-regular structures.

We need to lift an almost complex structure from the manifoldX to its tangent bundleT X. In local coordinates the liftJ^c is defined by

J^c =





J_i^h 0

t^a∂_aJ_i^h J_i^h



,

where t^a are coordinates in the tangent space. This lift is invariantly defined, see [7] for more details. After that we can use the induction onk.

Really, if J ∈ C^1,α then J_n^c ∈ C^α. Further we lift J-holomorphic map u: ∆→X toJ^c-holomorphic mapu^c: ∆→T X. This lift is defined as (3.4) u^c(ζ) = (u(ζ), du(ζ)(e1)),

wheree₁= (1,0).

Now remark that ifW is aJ-totally real submanifold inX then T W is aJ^c-totally real submanifold inT X. Really, letv∈T(T W)∩J^c(T(T W)).

Ifv = (v1, v2)in the trivialisation T(T X) = T X⊕T X then v1 ∈T W ∩ J(T W), implying that v₁ = 0. Hence v₂ ∈ T W ∩J(T W), implying that v2= 0. Thereforev= 0.

Further the lift u^c : ∆⁺ → T X of a J-holomorphic mapu: ∆⁺ → X with boundary values inW has its boundary values in T W (this is clear

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from the formula (3.4)). Applying Lemma 3.2 tou^c andT W we prove that the first derivative ofuwith respect toξ is of classC^1,α on∆⁺∪β. Here ζ=ξ+iη. TheJ^c-holomorphicity equation

(3.5) ∂u^c

∂η =J^c(u^c)∂u^c

∂ξ

implies that ^∂u_∂η^c is also of classC^1,α on∆⁺∪β. Therefore u∈ C^2,α up to β. By induction we conclude thatu is of classC^k+1,α up to β ifJ ∈ C^k,α andW is of classC^k+1,α.

Theorem 1.3 is proved.

4. Riemann-Hilbert Boundary-Value Problem

In this Section we develop one of the main tools of this paper - a sort of a Riemann-Hilbert problem. This will be used in the proof of Theorem 1.1.

Denote byS⁺={e^iθ :θ∈]0, π[}the upper semi-circle.

For the local considerations of the present Section we suppose that X = Cⁿ and W = iRⁿ = {z = x+iy : x = 0} and that J ∈ C^1,α is a small deformation ofJst. Fix also aJst-holomorphic mapu⁰: ∆−→Cⁿ of classL^1,p(∆),p >2, such thatu⁰(S⁺)⊂iRⁿ (so thatu⁰ extends holo- morphically to a neighborhood of S⁺ by the classical Schwarz Reflection Principle). ForJ close enough to Jst we will establish the existence and uniqueness of a J-holomorphic disc u close enough to u⁰ satisfying the boundary conditionu(S⁺)⊂iRⁿ.

Therefore for J close enough toJ_stwe study the solutions of (2.2) satisfying the boundary condition

Re u|S⁺ =0.

(4.1)

Denote byT₀^1,p the Banach space of (Rⁿ -valued) functionsϕ∈T^1,p vanishing on S⁺. This space is equipped with the standard norm k ϕ k_T1,p. Set nowϕ⁰ :=Re u⁰|S and remark that ϕ⁰ ∈ T₀^1,p because Re u⁰|S⁺ =0.

We replace the condition (4.1) for the solutions of the partial differential equation (2.2) by the condition

Re u|S=ϕ, (4.2)

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whereϕ∈T₀^1,p. Therefore we consider the following boundary-value problem











u_ζ−A_J(u)u_ζ = 0, Re u|S=ϕ, Im u(0) =a, (4.3)

for the given initial dataϕ∈T₀^1,p,a∈Rⁿ.

Lemma 4.1. — IfJis close enough toJ_stinC^1,α-norm then there exists a neighborhood U of ϕ⁰ in T₀^1,p, a neighborhood U⁰ of a⁰ := Im u⁰(0) in Rⁿ and a neighborhoodV ofu⁰ inL^1,p(∆)such that for eachϕ∈U and a∈U⁰ the boundary problem (4.3) admits a unique solutionu∈V.

Proof. — Consider the operator

LJ:L^1,p(∆)−→L^p(∆)×T^1,p×Rⁿ defined by

LJ :u7→





u_ζ−AJ(u)u_ζ Re u|S

Im u(0)



.

L_J smoothly depends on the parameter J. Denote by L˙_J(u) the Fréchet derivative of LJ at u. L˙J is continuous on the couple (J, u) and at Jst- holomorphicu⁰the derivative L˙_J_st(u⁰)is particularly simple:

L˙_J_st(u⁰) :L^1,p(∆)−→L^p(∆)×T^1,p×Rⁿ

L˙J_st(u⁰) : ˙u7→





˙ u_ζ Re ˙u|S

Im ˙u(0)



.

Let’s see that L˙J_st(u⁰) is an isomorphism. Indeed, given h∈ L^p(∆), ψ ∈ T^1,p anda∈Rⁿ then the function

˙

u=T_∆^CGh−iIm T^CG_∆ h(0)

+ia+T^SW(ψ−Re(T^CG_∆ h)|S) is of classL^1,p(∆)and satisfies the equation

L˙_J_st(u⁰)( ˙u) =



 h ψ a



.

Uniqueness of u˙ is obvious. Therefore by the Implicit Function Theorem everyLJ is aC¹-diffeomorphism of neighborhoods ofu⁰ inL^1,p(∆)and of (0, ϕ₀, a₀)inL^p(∆)×T^1,p×Rⁿ. SinceT₀^1,pis a closed subspace ofT^1,pthe

Lemma 4.1 follows.

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Let us formulate a corresponding statement in Hölder classes. Letk>1.

Fix a Jst-holomorphic map u⁰ : ∆ −→ Cⁿ of class C^k+1,α(∆) such that u⁰(S⁺)⊂iRⁿ. For every positive integerk denote byC₀^k,α(S)the Banach space of (Rⁿ -valued) functions ϕ∈ C^k,α(S)vanishing on S⁺. This space is equipped with the standard normk ϕ k_Ck,α(S). Set nowϕ⁰ :=Re u⁰|S.

ϕ⁰∈ C₀^k+1,α(S)becauseRe u⁰|_S+=0. Therefore we consider the boundary value problem (4.3) for the given initial dataϕ∈ C₀^k+1,α(S),a∈Rⁿ.

Lemma 4.2. — Supposek>1. IfJ is close enough to J_st inC^k,α-norm then for every16l6k:

(i) there exists a neighborhoodU ofϕ⁰ in C₀^l+1,α(S), a neighborhood U⁰ofa⁰:=Imϕ⁰(0)inRⁿand a neighborhoodV ofu⁰inC^l+1,α( ¯∆) such that for eachϕ∈U anda∈ U⁰ the boundary problem (4.3) admits a unique solutionu∈V;

(ii) the unit disc∆can be replaced in part (i) of the present Lemma by any bounded simply connected domain Ωwith C^∞ boundary and S⁺ can be replaced by any non-empty open arc.

Proof. — The part (ii) follows from (i) by the Riemann mapping theorem and the classical theorems on the boundary regularity of conformal maps.

For the proof of part consider the operator

LJ:C^l+1,α( ¯∆)−→ C^l,α( ¯∆)× C^l+1,α(S)×Rⁿ,

defined as in the proof of Lemma 4.1, but this time in other smoothness classes. One can literally repeat the arguments used there also in this case.

5. Reflection Principle-II: Real Analytic Case

Let us turn now to the proof of Theorem 1.1. We shall proceed in two steps.

5.1. Small deformations of the standard structure

Here we consider the case when J is a small real analytic deformation of J_st and W = iRⁿ. First we introduce suitable Banach spaces of real analytic functions using the complexification.

Denote by ∆²= ∆×∆ the standard bidisc inC². We define the space C_ω^1,α(∆)consisting of functionsu(orCⁿ-valued maps) of classC^1,α( ¯∆)with the following properties:

(i) uis a sum of a power seriesu(ζ) =P

kluklζ^kζ^lforζ∈∆.

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(ii) The “polarization” uˆ of u defined by u(ζ, ξ) =ˆ P

klu_klζ^kξ^l is a function holomorphic on∆² and of classC^1,α(∆²).

(iii) The mixed derivative _∂ζ∂ξ^∂²^u^ˆ is of classC^α(∆²).

We define the norm ofuas following:

kuk_C^1,α

ω (∆)=kˆuk_C_1,α_(∆²₎+

∂²uˆ

∂ζ∂ξ C^α(∆²)

.

Sinceu is the restriction ofuˆ onto the totally real diagonal {ξ = ¯ζ}, the polarizationˆuis uniquely determined byuand thereforeC_ω^1,α(∆)equipped with this norm is a Banach space.

Remark 5.1. — One has the following continuous inclusionO^1,α(∆)⊂ C_ω^1,α(∆): for u ∈ O^1,α(∆) the corresponding uˆ is simply u(ζ, ξ) =ˆ u(ζ).

Really, for suchuˆ one has _∂ζ∂ξ^∂²^ˆ^u = 0.

We denote byC_ω^1,α(∂∆⁺)the space of real functionsϕon∂∆⁺such that there exists a functionv∈ O^1,α(∆)satisfying the conditionRe v|_∂∆+ =ϕ.

In particular such functionϕ is real analytic on the interval(−1,1). The holomorphic functionvis unique up to an imaginary constant so we always assume thatIm v(0) = 0. We define the norm of ϕas a C^1,α norm of the corresponding functionv on∆. Then¯ C^1,α_ω (∂∆⁺)equipped with this norm, is a Banach space.

Furthermore, denote byC^1,α(∂∆⁺)the space of real continuous functions on∂∆⁺ which are of class C^1,α on the closed upper semi-circle and on the interval [−1,1]. Finally we denote by C₀^1,α(∂∆⁺) the space of real functions of class C^1,α(∂∆⁺) vanishing on the interval [−1,1]. The following statement is a consequence of the reflection principle.

Lemma 5.2. — For every function ϕ ∈ C₀^1,α(∂∆⁺) there exists u ∈ O^1,α(∆) such thatRe u|_∂∆+ =ϕ. In particular, the space C₀^1,α(∂∆⁺) is a subspace ofC_ω^1,α(∂∆⁺).

Proof. — Letϕ be a function of classC₀^1,α(∂∆⁺). Solving the Dirichlet problem forϕin the upper semi-disc, we obtain a harmonic functionhin

∆⁺ continuous on∆⁺ such that h|_∂∆+ =ϕ. Since hvanishes on[−1,1]it extends harmonically on∆by the classical reflection principle for harmonic functions. Namely, its extensionh^∗ is defined byh^∗(ζ) = −h(ζ) for ζ in the lower semi-disc∆⁻. Thus we obtain a function ˜hharmonic on∆ and continuous on ∆. Since the restriction ϕ of h on the closed upper semi- circle is a function of classC^1,α, it follows easily by the definition of the reflection h^∗ that the restriction ϕ˜ := ˜h|∂∆ of ˜h on ∂∆ is a function of

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classC^1,α(∂∆). Then the Schwarz integralT^SWϕ˜gives by Proposition 2.2 a function of classO^1,α(∆)whose real part coincides with˜h.

Lemma 5.3. — Ifu∈ C_ω^1,α(∆)thenRe u|_∂∆+∈ C_ω^1,α(∂∆⁺).

Proof. — Letu(ζ, ξ) =ˆ Pu_klζ^kξ^l be the polarization of uholomorphic in the bidisc∆² (that isu(ζ) = ˆu(ζ, ζ)). Then the functionh(ζ) = ˆu(ζ, ζ) is of class O^1,α(∆) and h|_[−1,1] = u|_[−1,1]. Denote b ϕ the restriction of Re(u−h)to∂∆⁺. Thenϕ∈ C^1,α₀ (∂∆⁺)and by Lemma 5.2 there exists a functionv∈ O^1,α(∆)such thatRe v|∂∆⁺ =ϕ. Since the functionh+v is of classO^1,α(∆), its real part gives the desired extension of the function

Re u|∂∆⁺.

We suppose everywhere below that our almost complex structureJ (and thereforeAJ in the equation for J holomorphic curves) is a real analytic matrix-valued function given by a convergent power seriesPaklz^kz^l with the radius of convergence big enough. The equation (2.2) on ∆ can be rewritten in the form

(5.1) (u−T_∆^CGAJ(u)u_ζ)_ζ = 0,

whereT_∆^CGdenotes the Cauchy - Green transform on∆. Define the map ΦJ :C^1,α( ¯∆)→ C^1,α( ¯∆),

as

(5.2) ΦJ:u7→u−T_∆^CGAJ(u)u_ζ.

Equation (5.1) means thatuisJ-holomorphic if and only ifΦJuis holomorphic with respect to Jst. The following lemma explains the choice of smoothness classes in this Section and is the principal step in the proof of Theorem 1.1.

Lemma 5.4. — ForJ close to Jst the operator ΦJ establishes a diffeo- morphism of neighborhoods of zero in the spaceC_ω^1,α(∆).

Proof. — First we prove thatΦJmaps the spaceC_ω^1,α(∆)to itself. Given functionu∈ C^1,α_ω (∆)denote the functionA_J(u)u_ζ byh. We need to prove thatT_∆^CGhbelongs toC_ω^1,α(∆). Consider the polarization ˆh(ζ, ξ) = ˆh(ζ, ξ) ofh. By Proposition 2.1 we have the representation

(5.3) T_∆^CGh(ζ) = ˆH(ζ, ζ)− 1 2πi

Z

∂∆

H(τ, τ)dτˆ τ−ζ . where

(5.4) H(ζ, ξ) =ˆ

Z

[0,ξ]

ˆh(ζ, ω)dω

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is a primitive of ˆhwith respect to ξ. Let’s study the primitive (5.4) of ˆh first. We point out that the functionˆhis of class C^0,α(∆²). Furthermore, the condition (iii) of the definition of the spaceC_ω^1,α(∆)implies that∂ˆh/∂ζ is of classC^0,α(∆²). Now the derivation of the integral (5.4) with respect toζ andξgives thatHˆ satisfies conditions (i), (ii), (iii) of the definition of the spaceC_ω^1,α(∆).

By Proposition 2.2 the Cauchy integral in the right hand side of (6.3) represents a function of class O^1,α(∆) and so also belongs to the space C_ω^1,α(∆).

Thus we obtain that ΦJ(u)belongs toC^1,α_ω (∆). Since the Fréchet derivative ofΦJ with respect touatu= 0andJ =Jstis the identity map, the lemma follows from the inverse mapping theorem.

Hence ΦJ is a diffeomorphism between neighborhoods of zero in the manifolds ofJ-holomorphic andJ_st-holomorphic maps of classC_ω^1,α(∆). In particular,J-holomorphic maps form a Banach submanifold inC_ω^1,α(∆)in a neighborhood of zero. We denote this manifold asO^1,α_ω,J(∆).

Remark 5.5. — Note that O^1,α_ω,J

st(∆) = O^1,α(∆) and that O^1,α_ω,J(∆) = ΦJ(O^1,α(∆)).

We use the notationO_ω,J,0^1,α (∆)for the submanifold of suchu∈ O^1,α_ω,J(∆) thatRe u|_[−1.1] ≡0. Its diffeomorphic image underΦJwe denote asMJ,0:=

ΦJ

O^1,α_ω,J,0(∆)

. ByR∂∆⁺denote “taking real part and restriction to∂∆⁺” operator. One has the following commutative diagram:

(5.5)

C_ω^1,α(∆) ⊃ O^1,α(∆) ^Φ

−1

−→J O^1,α_ω,J(∆)

↑i ↑i

C_ω^1,α(∆) ⊃ MJ,0 Φ⁻¹_J

−→ O^1,α_ω,J,0(∆) ^R−→^∂∆+ C₀^1,α(∂∆⁺) , where both i-s are natural imbeddings. For an unknown map v from O_ω,J,0^1,α (∆) and givenϕ∈ C^1,α₀ (∂∆⁺)consider the system

(R_∂∆+v=ϕ, Im v(0) =a.

(5.6)

Fix a Jst-holomorphic map v⁰ ∈ O^1,α(∆) such thatRe v⁰|_[−1,1] ≡0 and setϕ⁰=Re v⁰|∂∆⁺.

Lemma 5.6. — For real analyticJclose enough toJstinC^1,α-norm there exists a neighborhoodU ofϕ⁰ in C₀^1,α(∂∆⁺), a neighborhoodU⁰ ofa⁰ :=

Im v⁰(0)inRⁿ and a neighborhoodV ofv⁰inC_ω^1,α(∆)such that forϕ∈U anda∈U⁰ the system (5.6) admits a unique solutionv∈V ∩ O^1,α_ω,J,0(∆).

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Proof. — Setting v = Φ⁻¹_J w we replace (5.6) by the boundary value problem for an unknown mapw∈ MJ,0, i.e., we shall write it as

(R∂∆⁺Φ⁻¹_J w=ϕ, Im Φ⁻¹_J w(0) =a.

(5.7)

At J = J_st that Φ_J = Id and M_J_st_,0 = O^1,α₀ (∆) := {w ∈ O^1,α(∆) : Re w|[−1,1]≡0}. The surjectivity condition for the operator obtained by the linearization of (5.7) atw⁰=v⁰ and J =Jst and of the operatorΦ⁻¹_J is reduced to the resolution of the following system

(5.8)

(R_∂∆+w˙ =ψ, Im ˙w(0) =a,

for an arbitrary given function ψ ∈ C₀^1,α(∂∆⁺), arbitrarya ∈Rⁿ and an unknown mapw˙ ∈ O^1,α₀ (∆). By Lemma 5.2 we obtain a solution for any given right hand side of (5.8). The uniqueness ofw˙ is obvious. Therefore the linearization of (5.7) is a bĳective operator atJ =Jst. By continuity it will be bĳective fromT₀M_J,0 toC₀^1,α(∂∆⁺)⊕Rⁿ also forJ close toJ_st. Now the Implicit Function Theorem implies the desired statement.

5.2. General case

Now we prove Theorem 1.1. According to Theorem 1.3 (which we will prove in the next section) the mapuisC^∞smooth up to the arcβ. We replace the unit disc by the upper semi-disc∆⁺andβby the interval(−1,1).

By the classical results on the interior regularity of pseudo-holomorphic maps we can assume that the mapuis real analytic in a neighborhood of

∆¯⁺\[−1,1]. Furthermore, the statement is local so shrinking∆⁺ if necessary we assume thatuis of classC^∞(∆⁺). We can also assume that X is the unit ball ofCⁿ equipped with a real analytic almost complex structure J withJ(0) =J_st and that W =iRⁿ andu(0) = 0.

Since our considerations are local, we can reduce them to the case of a small deformation of the standard complex structure. Indeed, our map u admits the expansion u(ζ) = bζ +o(|ζ|) near the origin. For t > 0 consider the real analytic structuresJt(z) =J(tz). They tend to Jst as t tends to0. Mapsu^t(ζ) =t⁻¹u(tζ)areJ_t-holomorphic and tend to the map u⁰:ζ−→bζ as t−→0 which is viewed as aJst-holomorphic map.

For t small enough let v^t be the solution of the boundary-value problem (5.6) withϕ^t:=Re u^t|∂∆⁺given by Lemma 5.6. This solution is unique

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in the class of solutions real analytically extendable past(−1,1). However, this still does not give the desired analyticity of u^t since this boundary- value problem could admit solutions in other smoothness classes of maps.

So we need the uniqueness statement of Lemma 4.1. LetH : ∆→∆⁺be a biholomorphic map fixing the points−1and1. ThenH is of classC^1/2(∆) and extends analytically through the open upper and lower semi-circles. In particular,H ∈L^1,p(∆) for anyp <4. Since v^t ∈ C^1,α(∆⁺) the composi- tionv^t◦H is inL^1,p(∆). At the same timeϕ^t◦H belongs toT^1,pfor every p <4. Applying the uniqueness statement of Lemma 4.1 we conclude that v^t◦H = u^t◦H which implies that the maps u^t are real analytic up to (−1,1)fort small enough. This proves finally thatuextends as a real analytic map past(−1,1). Since it satisfies the real analytic condition (2.1) on an open set, the extension is a J-holomorphic map. This proves the Theorem 1.1.

6. Compactness

6.1. Compactness theorem: definitions

We start with recalling some notions and definitions relevant to the formulation of Gromov compactness theorem as it is stated in [9, 10]. For a more detailed exposition we refer to these papers.

1. Structures.We fix a Riemannian manifold(X, h), a compact subset K bX and a sequence of almost-complex structuresJ_n of class C^k,α on X, which converge on K inC^k,α-topology to an almost-complex structure J,k>0,0< α <1. The latter is supposed to be defined on the wholeX. Areas ofJn-complex curves will be measured with respect to the “hermiti- zations”hJ_n(·,·) := ¹₂(h(·,·) +h(Jn·, Jn·))ofh(which converge tohJ).

2. Curves.We are given a sequence {C_n} of nodal curves with bound- ary, parametrized by the same compact, oriented real surface(Σ, ∂Σ)with boundary. We suppose that some parametrizationδn : ¯Σ→C¯n are given as well as some Jn-holomorphic maps un : Cn → X such that (Cn, un) becomes astable curve over (X, Jn), see Definitions 2.1 - 2.3 from [10].

3. Boundedness of areas.We suppose that h_J_n-areas of u_n(C_n) are uniformly bounded and thatun(Cn)⊂K for alln.