Wong-Rosay Theorem in almost complex manifoldsH. Gaussier, A. SukhovTo cite this version:H. Gaussier, A. Sukhov. Wong-Rosay Theorem in almost complex manifolds. date de redaction:2003-6-18. 2004. HAL Id: hal-00000526https://hal.archives-ouvertes.fr/hal-00000526v2Submitted on 26 Feb 2004HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.ccsd-00000526 (version 2) : 26 Feb 2004WONG-ROSAY THEOREM IN ALMOST COMPLEX MANIFOLDSHERVE´ GAUSSIER AND ALEXANDRE SUKHOVAbstract. We study the compactness of sequences of diffeomorphisms in almost complexmanifolds in terms of the direct images of the standard integrable structure.IntroductionThe classical Wong-Rosay theorem states that every domain in the euclidean space Cnor more generally in a complex manifold of dimension n, with an automorphism orbit ac-cumulating at a C2 strictly pseudoconvex point, is biholomorphically equivalent to the unitball in Cn (see [4, 8, 9, 11]). The aim of this paper is to extend this theorem to strictlypseudoconvex domains in almost complex manifolds. Our main result can be considered asa compactness theorem for sequences of diffeomorphisms. This shows that the convergenceof such sequences can be controlled in terms of the direct images of the standard complexstructure. Our approach is based on the scaling method, introduced by S.Pinchuk [8] for thecase of the integrable structure. In order to apply it in the almost complex case, we needsubstantial modifications. In particular we use lower estimates for the Kobayashi infinitesimalpseudometric on almost complex manifolds [5] and a priori estimates for J-holomorphic curves[10].1. Statement of the resultsAn almost complex manifold (M,J) is a smooth (C∞) real manifold equipped with analmost complex structure J , that is a C∞-field of complex linear structures on the tangentbundle TM ofM . Given two almost complex manifolds (M,J) and (M ′, J ′) and a smooth mapf fromM ′ toM we say that f is (J ′, J)-holomorphic if its differential df : TM ′ → TM satisfiesdf ◦ J ′ = J ◦ df on TM . We denote by O(J′,J)(M ′,M) the set of (J ′, J)-holomorphic mapsfromM ′ toM and by Diff(J′,J)(M′,M) the set of (J ′, J)-holomorphic diffeomorphisms fromM ′ to M . The set Diff(J′,J)(M′,M) is generically empty. However given a diffeomorphismf from M ′ to M and an almost complex structure J ′ on M ′ then f ∈ Diff(J,J′)(M,M ′)for the almost complex structure J = df ◦ J ′ ◦ df−1 naturally associated with f . If Mand M ′ are two domains in Cn, if J = J ′ = Jst, the usual standard structure on Cn, andf ∈ Diff(Jst,Jst)(M ′,M) we simply say that f is a biholomorphism from M ′ to M , or thatM is biholomorphic to M ′.We consider the following situation :• D is a bounded domain in C2,• p is a point in a four dimensional almost complex manifold (M,J), U is a relatively compactneighborhood of p in M , r is a C2 strictly J-plurisubharmonic function in a neighborhood ofU¯ satisfying r(p) = 0 and dr 6= 0 on U ,• (rν)ν is a sequence of C2 functions in a neighborhood of U¯ such that limν→∞ ‖rν−r‖C2(U¯) =0,• for every ν, fν is a diffeomorphism from D to Gν ⊂M with Gν ∩ U = {x ∈ U : rν(x) < 0}• for every ν, the almost complex structure Jν := dfν ◦ Jst ◦ (dfν)−1 extends smoothly to U¯ .Then we have :Date: 2004-2-26.2000 Mathematics Subject Classification. 32H02, 53C15.12 HERVE´ GAUSSIER AND ALEXANDRE SUKHOVTheorem 1. Assume that limν→∞ ‖Jν − J‖C2(U¯) = 0. If there is a point x0 ∈ D such thatlimν→∞ fν(x0) = p, then D is biholomorphic to the unit ball B2 in C2.Theorem 1 implies the following compactness result :Corollary 1. Let D be a bounded domain in C2, not biholomorphic to the unit ball B2 andlet G = {x ∈M : r(x) < 0} be a relatively compact domain in (M,J), where r is a C2 strictlyJ-plurisubhamonic function on M . Let (fν)ν be a sequence of diffeomorphisms from D to G,defined in a neighborhood of D¯. If ‖Jν − J‖C2(D¯) →ν→∞ 0 then (fν)ν is compact.Finally we have an analogue of the Wong-Rosay theorem in almost complex manifolds :Theorem 2. Let D = {x ∈ M : r(x) < 0} be a relatively compact domain in a four dimen-sional almost complex manifold (M,J), where r is a C2 strictly plurisubharmonic functionon M . Assume that the set Diff(J,Jst)(D,B2) is empty. Then the set Diff(J,J)(D,D) iscompact for the compact-open topology.2. PreliminariesThe following Lemma shows that every almost complex structure J on an almost complexmanifold may be represented locally at p ∈ M as a small C2 deformation of the standardstructure (see [5]).Lemma 1. For every λ0 > 0 there exist a neighborhood U0 of p and a coordinate diffeomor-phism z : U0 → B2 such that z(p) = 0, dz(p) ◦ J(p) ◦ dz−1(0) = Jst and the direct imageJˆ = z∗(J) satisfies ||Jˆ − Jst||C2(B¯2) ≤ λ0.Proof. Shrinking U if necessary there exists a diffeomorphism z from U onto B2 satisfyingz(p) = 0 and dz(p) ◦ J(p) ◦ dz−1(0) = Jst. For λ > 0 consider the dilation dλ : t 7→ λ−1tin C2 and the composition zλ = dλ ◦ z. Then limλ→0 ||(zλ)∗(J) − Jst||C2(B¯2) = 0. SettingU = z−1λ (B2) for λ > 0 small enough, we obtain the desired statement. In the sequel we will consider the diffeomorphism z given by Lemma 1, for sufficientlysmall λ0.2.1. Plurisubharmonic functions. Let (M,J) be an almost complex manifold. We denoteby TM the real tangent bundle of M and by TCM its complexification. If T(1,0)M := {X ∈TCM : JX = iX} = {ζ− iJζ, ζ ∈ TM}, T (0,1)M := {X ∈ TCM : JX = −iX} = {ζ+ iJζ, ζ ∈TM} then TCM = T (1,0)M ⊕ T (0,1)M . The set of complex forms of type (1, 0) on M isdefined by T(1,0)M = {w ∈ T ∗CM : w(X) = 0, ∀X ∈ T (0,1)M} and the set of complexforms of type (0, 1) on M by T(0,1)M = {w ∈ T ∗CM : w(X) = 0, ∀X ∈ T (1,0)M}. ThenT ∗CM = T(1,0)M ⊕ T(0,1)M and the operators ∂J and ∂¯J are defined on the space of smoothcomplex functions on M by ∂Ju = du(1,0) ∈ T(1,0)M and ∂¯Ju = du(0,1) ∈ T(0,1)M , for everycomplex smooth function u on M .Let r be a C2 function on M . The Levi form of r is defined on TM byLJ(r)(X) := −d(J⋆dr)(X, JX).We recall that an upper semicontinuous function u on (M,J) is called J-plurisubharmonic onM if the composition u ◦ f is subharmonic on ∆ for every f ∈ OJ(∆,M). Then we have thefollowing characterization of J-plurisubharmonic functions (see [5]) :Proposition 1. Let u be a C2 real valued function on M . Then u is J-plurisubharmonic onM if and only if LJ(u)(X) ≥ 0 for every X ∈ TM .Hence, following [5], we say that a C2 real valued function u on M is strictly J-plurisubharmonic on M if LJ(u) is positive definite on TM .We point out that the strict J-plurisubharmonicity is stable with respect to small C2deformations of the almost complex structure and of the function.WONG-ROSAY THEOREM IN ALMOST COMPLEX MANIFOLDS 32.2. Kobayashi-Royden pseudometric. We denote by ∆ the unit disc in C. Let (M,J)be an almost complex manifold. According to [7] for every p ∈ M there is a neighborhoodV of 0 in TpM such that for every v ∈ V there exists f ∈ OJ (∆,M) satisfying f(0) = p,df(0)(∂/∂x) = v. We may therefore define the Kobayashi-Royden infinitesimal pseudometricK(M,J) and the integrated pseudodistance dK(M,J) (the upper semicontinuity of K(M,J) on thetangent bundle TM of M is proved in [6]) :Definition 1. (i) For p ∈ M and v ∈ TpM , K(M,J)(p, v) is the infimum of the set ofpositive α such that there exists a J-holomorphic disc f : ∆ → M satisfying f(0) = p anddf(0)(∂/∂x) = v/α.(ii) Let p, q ∈ M . Denote by Γp,q the set of all C1-paths γ : [0, 1] → M satisfying γ(0) =p, γ(1) = q. Then dK(M,J)(p, q) = infγ∈Γp,q∫ 10 K(D,J)(γ(t), γ′(t))dt.As in the complex case, the integrated pseudodistance is decreasing under the action of(J ′, J)-holomorphic maps. Hence if f ∈ Diff(J′,J)(M ′,M) then the inverse map f−1 is inDiff(J,J′)(M,M′) and we have for every p′, q′ ∈M ′ :(1) dK(M,J)(f(p′), f(q′)) = dK(M ′,J′)(p′, q′).The following results are proved in [5] :Proposition A. (Localization principle) Let D be a domain in an almost complex manifold(M,J), let p ∈ D¯ and let U be a neighborhood of p in M (not necessarily contained in D). Letu be a C2 function on D¯, negative and J-plurisubharmonic on D. We assume that −L ≤ u < 0on D∩U and that u−c|z|2 is strictly J-plurisubharmonic on D∩U , where c and L are positiveconstants and z is the diffeomorphism given by Lemma 1. Then there exist a positive constants and a neighborhood V ⊂⊂ U of p, depending on c and L only, such that for q ∈ D ∩ V andv ∈ TqM we have K(D,J)(q, v) ≥ sK(D∩U,J)(q, v).The next Proposition gives a lower bound on the Kobayashi-Royden infinitesimal pseudo-metric (more precise lower estimates are given in Theorem 1 of [5]).Proposition B. Let D be a relatively compact domain in an almost complex manifold (M,J).Let u be a C2 function on D¯, satisfying −L ≤ u < 0 on D and u − c|z|2 is strictly J-plurisubharmonic on D, where c and L are positive constants and z is given by Lemma 1.Then there exist positive constants C and λ0, depending on c and L only, such that for everyalmost complex structure J ′ defined in a neighborhood of D¯ and such that ‖J ′− J‖C2(D¯) ≤ λ0we have : K(D,J′)(p, v) ≥ C‖v‖ for every p ∈ D and every v ∈ TpM .Finally we have the boundary behaviour of the Kobayashi pseudodistance in a strictlyJ-pseudoconvex domain.Proposition C. Let D, Dν be domains in an almost complex manifold (M,J) and let p ∈ ∂D.Assume that there is a neighborhood U of p such that D ∩ U = {x ∈ U : r(x) < 0}, where ris a C2 strictly J-plurisubharmonic function in a neighborhood of U¯ . If Dν ∩ U = {x ∈ U :rν(x) < 0}, where rν is a sequence of C2 functions in a neighborhood of U¯ , converging to r inthe C2(U¯) convergence, then for every q ∈ D ∩ U there is ν0 such that for ν ≥ ν0 we have :limx∈Dνx→pdK(Dν ,J)(x, q) = +∞.3. Proof of theorem 1In this Section we assume that the assumptions of Theorem 1 are satisfied.3.1. Attraction property. The following Lemma is a direct application of Proposition C.Lemma 2. For every K ⊂⊂ D we have : limν→∞ fν(K) = p.4 HERVE´ GAUSSIER AND ALEXANDRE SUKHOVProof of Lemma 2. Let K ⊂⊂ D be such that x0 ∈ K. Since the function x 7→ dKD (x0, x)is bounded from above by a constant C on K, it follows from the decreasing property of theKobayashi pseudodistance that(2) dK(Gν ,Jν)(fν(x0), fν(x)) ≤ Cfor every ν and every x ∈ K. Moreover, r is strictly J-plurisubharmonic in a neighbor-hood of U¯ and the sequence (rν)ν converges to r in the C2(U¯) convergence. It follows fromProposition C that for every V ⊂⊂ U , containing p, we have :(3) limν→∞dK(Gν ,Jν)(fν(x0), Gν ∩ ∂V ) = +∞.It follows from conditions (2) and (3) that fν(K) ⊂ V for every sufficiently large ν. Thisgives the statement. According to [10] Corollary 3.1.2, there exist a neighborhood U of p in M and complexcoordinates z = (z1, z2) : U → B2 ⊂ C2, z(p) = 0 such that z∗(J)(0) = Jst and moreover, amap f : ∆→ B2 is J ′ := z∗(J)-holomorphic if it satisfies the equations∂fj/∂ζ¯ = Aj(f1, f2)(∂fj/∂ζ), for j = 1, 2,(4)where Aj(z) = O(|z|), j = 1, 2.In order to obtain such coordinates, one can consider two transversal foliations of B byJ ′-holomorphic curves (see [7]) and then take these curves into the lines zj = const bya local diffeomorphism. The direct image of the almost complex structure J under sucha diffeomorphism has a diagonal matrix J ′(z1, z2) = (ajk(z))jk with a12 = a21 = 0 andajj = i+ αjj where αjj(z) = O(|z|) for j = 1, 2.In what follows we omit the prime and denote this structure again by J . We also stilldenote by Jν the structure z∗(Jν) defined in B2 and we set pν := z(fν(x0)). Hence we mayassume that U = B2 and that Gν ∩U = Gν ∩B2 are domains in C2, contained in B2. Let G :={x ∈ B2 : r(x) < 0}. We may assume that the complex tangent space T0(∂G)∩J(0)T0(∂G) =T0(∂G) ∩ iT0(∂G) is given by {z2 = 0}. In particular, we have the following expansion forthe defining function r of G on B2 : r(z, z¯) = 2Re(z2) + 2ReK(z) +H(z, z¯) +O(|z|3), whereK(z) =∑λk,lzkzl, λk,l = λl,k and H(z, z¯) =∑λk,l¯zkz¯l, λk,l¯ = λ¯l,k¯.Lemma 3. The domain G is strictly Jst-pseudoconvex near the origin.Proof of Lemma 3. Consider a complex vector v = (v1, 0) tangent to ∂D at the origin. Let f :∆→ C2 be a J-holomorphic disc centered at the origin and tangent to v : f(ζ) = vζ+O(|ζ|2).Since A2 = O(|z|), it follows from the J-holomorphy equation (4) that (f2)ζζ¯(0) = 0. Thisimplies that (r ◦ f)ζζ¯(0) = H(v). Thus, the Levi form with respect to J coincides with theLevi form with respect to Jst on the complex tangent space of ∂G at the origin. For sufficiently large ν let qν be the unique point on ∂Gν ∩ B2 such that |qν − pν | =dist(pν , ∂Gν ∩ B2). Since the sequence (Jν)ν converges to J in the C2(B¯2) convergence,there exists a diffeomorphism Aν defined on B2 such that Aν(qν) = 0 and if Aν(z) =: zν =(zν1 , zν2 ) then the expansion of rν in the zν-coordinates is given by : rν = 2λνnRe(zν2 ) +2Re(∑2k,l=1 λνk,lzνkzνl )+∑2k,l=1 λνk,l¯zνk z¯νl +O(|zν |3), λνk,l = λνl,k, λνk,l¯ = λ¯νl,k¯, where the conditionO(|zν |3) is uniform with respect to ν. Finally the diffeomorphism Aν can be chosen such thatd(Aν) ◦ Jν ◦ d(Aν)−1 is represented by a diagonal matrix (aνjk(zν))jk with aν12 = aν21 = 0and aνjj = i + ανjj(zν) where aνjj converges to ajj , with its first derivatives, uniformly oncompact subsets of B2. We note that Aν converges to the identity map in any Ck(B¯2) norm.Since rν converges to r in the C2(B¯2) norm by assumption, it follows that : limν→∞ λν2 = 1,limν→∞λνk,l = λk,l and limν→∞ λνk,l¯= λk,l¯ for k, l = 1, 2.For every ν ≥ 1, let τν := dist(pν , qν). We define the dilation Λν : Aν(B2) → C2 byΛν(zν) = (zν1/√τν , zν2/τν), in the zν-coordinates. Let J˜ν be the almost complex structureWONG-ROSAY THEOREM IN ALMOST COMPLEX MANIFOLDS 5defined on Λν ◦Aν(B2) by J˜ν := Λν ◦ dAν ◦ Jν ◦ d(Aν)−1 ◦ (Λν)−1. Since Aν is (Jν , dAν ◦ Jν ◦d(Aν)−1)-holomorphic, Λν is (Jν , J˜ν)-holomorphic. Finally, we consider the domainG˜ν := Λν ◦Aν(Gν ∩ B2) = {z ∈ Λν ◦Aν(B2) : rν((Λν ◦Aν)−1(z), (Λν ◦Aν)−1(z)) < 0}.The next Lemma gives the limit behaviour of the domains G˜ν and of the almost complexstructures J˜ν .Lemma 4. The following conditions are satisfied :(i) limν→∞ G˜ν = G = {z ∈ C2 : Re(z2) + 2Re(K(z1, 0)) +H((z1, 0), (z¯1, 0)) < 0},(ii) limν→∞ J˜ν = Jst uniformly on compact subsets of C2.Proof of Lemma 4. The convergence in statement (i) is the Hausdorff convergence on compactsubsets. Condition (i) is a direct consequence of the convergence of (rν)ν to r and is similarto the usual complex case (see [8]).Proof of (ii). For every ν the almost complex structure J˜ν is represented by the diagonalmatrix (bνjk(z))jk where bνjj(z) = aνjj((Λν)−1(z)) = i + ανjj(√τνz1, τνz2) →ν→∞ i, uniformlyon compact subsets of C2. For every ν, let F ν := Λν ◦ Aν ◦ fν . Then F ν is a (Jst, J˜ν)-holomorphic map from(fν)−1(Gν ∩ B2) to G˜ν satisfying F ν(x0) = (0,−1) and we have :Proposition 2. (i) We may extract from (F ν)ν a subsequence converging, uniformly oncompact subsets of D, to a Jst-holomorphic map F : D −→ G¯,(ii) We may extract from ((F ν)−1)ν a subsequence converging, uniformly on compact sub-sets of G, to a Jst-holomorphic map F˜ : G −→ D¯,(iii) F is a biholomorphism from D to G with F−1 = F˜ .Statement (iii) is the conclusion of Theorem 1. Indeed it follows from Lemma 3 thatH((z1, 0), (z¯1, 0)) = αz1z¯1 with α > 0. Then the map (z1, z2) 7→ (√αz1, z2 + H(z1, 0)) is abiholomorphism from G to the unbounded representation of the unit ball H = {(z1, z2) ∈ C2 :Re(z2) + |z1|2 < 0}.Our proof of Proposition 2 is based on the method developped by F.Berteloot and G.Coeure´ [3]and by F.Berteloot [2]. We first prove the following Lemma :Lemma 5. There exist C0 > 0, δ0 > 0 and r0 > 0 such that for every 0 < δ < δ0, for everyν >> 1 and for every Jν-holomorphic disc gν : ∆→ Gν we have :gν(0) ∈ Q(0, δ)⇒ gν(r0∆) ⊂ Q(0, C0δ),where Q(0, δ) := {z ∈ C2 : |z1| <√δ, |z2| < δ}.Proof of Lemma 5. Assume by contradiction that there exist Cν → ∞, ζν → 0 in ∆ andJν -holomorphic discs gν : ∆ → Gν such that gν(0) ∈ Q(0, δν) and gν(ζν) 6∈ Q(0, Cνδν).Let dν : z 7→ (z1/√δν , z2/δν), let hν := dν ◦ gν and let Jν := dν(Jν). We set Hν :={z ∈ dν(U) : rν ◦ (dν)−1(z, z¯) < 0}. It follows from Lemma 4 part (ii) that ρν := rν ◦ dνconverges to ρ := Re(z2) + |z1|2, uniformly on compact subsets of C2 and Jν convergesto Jst, uniformly on compact subsets of C2. There exists A > 0 such that the functionρ+Aρ2 is strictly Jst-plurisubharmonic on Q(0, 5). Hence for sufficiently large ν the functionρν +A(ρν)2 is strictly Jν-plurisubharmonic on Q(0, 4). Moreover we can extend this functionas a Jν-plurisubharmonic function on Hν . In particular it follows from Proposition A andProposition B that there is a positive constant C such that K(Hν ,Jν)(z, v) ≥ C‖v‖ for everyz ∈ Hν ∩Q(0, 3), v ∈ C2. Therefore, there exists a constant C′ > 0 such that ‖ dhν(ζ) ‖≤ C′for any ζ ∈ (1/2)∆ satisfying hν(ζ) ∈ Hν ∩Q(0, 3). On the other hand, the sequence |hν(ζν)|tends to +∞. Denote by [0, ζν ] the segment (in C) joining the origin and ζν and let ζ′ν ∈ [0, ζν ]be the point closest to the origin such that hν([0, ζ′ν ]) ⊂ Hν ∩Q(0, 2) and hν(ζ′ν) ∈ ∂Q(0, 2).Since hν(0) ∈ Q(0, 1), we have |hν(0) − hν(ζ′ν)| ≥ C′′ for some constant C′′ > 0. Let6 HERVE´ GAUSSIER AND ALEXANDRE SUKHOVζ′ν = rνeiθν , rν ∈]0, 1[. Then|hν(0)− hν(ζ′ν)| ≤∫ rν0‖ dhν(teiθν ) ‖ dt ≤ C′rν −→ 0.This contradiction proves Lemma 5.We note that Lemma 5 is also satisfied replacing Jν -holomorphic discs by (Jst, Jν)-holomorphic maps. As a corollary we have the followingLemma 6. For any compact subset K ⊂ D the sequence of norms (‖F ν‖C0(K))ν is bounded.For the proof we can consider a covering of K by sufficiently small balls, similarly to [3],p.84. Indeed, consider a covering of K by the balls pj + r0B, j = 0, ..., N where r0 is givenby Lemma 5 and pj+1 ∈ pj + r0B for any j. Since limν→∞(Aν ◦ fν)(x0) = 0, we obtain, forν large enough, that Aν ◦ fν(p0 + rB) ⊂ Q(0, 2Cτν), then Aν ◦ fν(p1 + rB) ⊂ Q(0, 4C2τν).Continuing this process we obtain that Aν ◦ fν(pN + rB) ⊂ Q(0, 2NCNτν). Since τν → 0 andF ν = Λν ◦Aν ◦ fν we obtain Lemma 6.We prove now Proposition 2. Part (i). Lemma 6 implies that the sequence (F ν)ν is bounded(in the C0 norm) on any compact subset K of D. Covering K by small bidiscs, considertwo transversal foliations by holomorphic curves on every bidisc. Since the restriction ofF ν on every such curve is uniformly bounded in the C0-norm, it follows by the well-knownelliptic estimates that it is bounded in Cl norm for every l (see [10]). Since the bounds areuniform with respect to curves, this implies that the sequence (F ν)ν is bounded in everyCl-norm. So the family (F ν)ν is relatively compact by Ascoli theorem. Let F be a clusterpoint of (F ν)ν . We still denote by (Fν)ν an appropriate subsequence converging (uniformlyon compact subsets of D) to F . Passing to the limit in the holomorphy condition J˜ν ◦ dF ν =dF ν ◦ J˜ν , we obtain that F is holomorphic with respect to Jst.Part (ii). Let 0 < r < 1, B(0, r) = {z ∈ C2 : ‖z‖ < r} and let Φ be a biholomorphism fromG to B2 satisfying Φ((0,−1)) = (0, 0) (this is a biholomorphism for the standard structureJst on G and B2). According to Lemma 4 (ii) we have, for sufficiently large ν, the inclusionΦ−1(B(0, r)) ⊂ G˜ν . Since the sequence (J˜ν)ν converges uniformly on compact subsets of Gto Jst (in C2 norm) by Lemma 4 (iii), the sequence (J ′ν := dΦ ◦ J˜ν ◦ dΦ−1)ν converges toJst uniformly on compact subsets of B2 (in C2 norm). Fix 0 < r < 1. Then there exists apositive constant Cr and r < r′ < 1 such that for every z ∈ B(0, r) and every unitary vectorv in C2 there is a Jst-holomorphic disc ϕz,v : ∆ → B2 satisfying ϕz,v(0) = z, ϕ′(0) ∈ Cv,|ϕ′(0)| ≥ Cr and ϕz,v(∆(0, 1/2)) ⊂ B(0, r′). According to Section 5.4a of [7] (study of thestability of J-holomorphic discs under small smooth deformations of a given almost complexstructure J) there exists a positive contant C′r and r < r′′ < 1 such that for sufficiently largeν, for every z ∈ B(0, r) and for every unitary vector v ∈ C2, there is a J ′ν -holomorphic discfJ′ν ,z,v : ∆→ B2, centered at z, satisfying(5) d(fJ′ν ,z,v)(0)(∂/∂x) = αv with α ≥ C′rand fJ′ν ,z,v)(∆(0, 1/3)) ⊂ B(0, r′′). Since for sufficiently large ν the Jst-holomorphic discf˜J′ν ,z := (Φ ◦ F ν)−1 ◦ fJ′ν ,z satisfies the inclusion f˜J′ν ,z(∆) ⊂ D ⊂⊂ C2, there exists a positiveconstant C′ such that(6) |(f˜J′ν ,z)′(0)| ≤ C′for sufficiently large ν.It follows from conditions (5) and (6) that the first derivative of (Φ ◦ F ν)−1 is boundedfrom below by a positive constant on B(0, r), uniformly with respect to ν >> 1. By AscoliTheorem we may extract from ((F ν)−1)ν a subsequence that converges uniformly on B(0, r)to a holomorphic map F˜r : B(0, r) → D¯. In particular F˜r′ = F˜r on B(0, r) for r′ > r. Thisproves condition (ii).Part (iii). We know that F (x0) = 0 ∈ G. Assume now that there is x ∈ D such thatF (x) ∈ ∂G and let γ be a C1-path in D such that γ(0) = x0, γ(1) = x. We considerWONG-ROSAY THEOREM IN ALMOST COMPLEX MANIFOLDS 7t0 ≤ 1 such that F (γ(0, t0[)) ⊂ G and F (γ(t0)) ∈ ∂G. Since G is complete hyperbolic (G isbiholomorphic to the unit ball in C2), we obtain that limt→t0 dKH(0, F (γ(t))) =∞. However,for every t < 1 :dKG (0, F (γ(t))) ≤ sups∈[0,1]dKD (x0, γ(s)) <∞by the compactness of γ([0, 1]) in D. This is a contradiction, implying that F (D) ⊂ G. Weprove now that F˜ (G) ⊂ D. Since D is bounded in C2, D is hyperbolic for the standardstructure Jst. Let r > 0 and consider the Kobayashi ball BK(D,Jst)(x0, r). Since fν is abiholomorphism from (D, Jst) to (Gν , Jν), we have fν(BK(D,Jst)(x0, r)) = B(Gν ,Jν)(fν(x0), r)(see equality (1)). Since the sequence of points (fν(x0))ν converges to p we obtain fromProposition C that B(Gν ,Jν)(fν(x0), r) ⊂⊂ Gν ∩U for sufficiently large ν. In particular, since(fν)−1 is continuous on Gν , we have : BK(D,Jst)(x0, r) ⊂⊂ D. This implies that D is completehyperbolic. Assume that q ∈ G is such that F˜ (q) ∈ ∂D. Then there exists r > 0 such thatq ∈ BK(G,Jst)(0, r). Let γ : [0, 1]→ G be a C1-path such that γ(0) = 0, γ(1) = q and γ([0, 1]) ⊂BK(G,Jst)(0, 2r). We define t0 ≤ 1 such that F˜ (γ([0, t0[)) ⊂ D and F˜ (γ(t0)) ∈ ∂D. By thecomplete hyperbolicity of D, we have : limt→t0 dK(D,Jst)(x0, F˜ (γ(t))) = ∞, which contradictsthe condition dK(D,Jst)(x0, F˜ (γ(t))) ≤ d(G,Jst)(0, γt) ≤ 2r for every t < t0. Consequently :F˜ (G) ⊂ D. Now, since for every ν, (F ν)−1 ◦ F ν = idD and F ν ◦ (F ν)−1 = idG we obtain,by taking the limit when ν → ∞, that F˜ ◦ F = idD and F ◦ F˜ = idG, meaning that F is abiholomorphism from D to G. This proves Theorem 1. Corollary 1 is a direct consequence of Theorem 1.4. Proof of Theorem 2We explain now how to adapt the proof of Theorem 1 to obtain Theorem 2. Hence weassume in this Section that the assumptions of Theorem 2 are satisfied.We have the followingLemma 7. For every 0 ∈ K ⊂⊂ ∆ we have : limν→∞(fν ◦f)(K) = p, uniformly with respectto f ∈ O(Jst,J)(∆, D) such that f(0) = x0.Proof of Lemma 7. The proof follows line by line the proof of Lemma 2, replacing fν by fν ◦fwhere f ∈ O(Jst,J)(∆, D), f(0) = x0. Indeed since the function x 7→ dK(∆,Jst)(0, x) is boundedfrom above on K by a positive constant C, we have :dK(D,J)(fν ◦ f(0), fν ◦ f(x)) ≤ Cfor every ν ≥ 1 and for every x ∈ K. Fix a neighborhood V ⊂⊂M of p. Sincelimν→∞dK(D,J)(fν(x0), D ∩ ∂V ) =∞we have fν ◦ f(K) ⊂ V for sufficiently large ν, uniformly with respect to f ∈ O(Jst,J)(δ,D)such that f(0) = x0. Proof of Theorem 2. Assume by contradiction that there is a sequence (fν)ν inDiff(J,J)(D,D) and points x0 ∈ D, p ∈ ∂D such that limν→∞ fν(x0) = p. Lemma 7implies that : limν→∞ fν(K) = p, for every compact K in D.Let U , z and Aν defined as in page 5 and let Λν is the dilation defined page 6. We considerGν := D, Jν := J and J˜ν := Λν ◦ d(Aν) ◦ J ◦ d(Aν)−1 ◦ (Λν)−1 for every ν. We may assumeas in the proof of Theorem 1 that D ∩ U is in C2 and we still denote by J the associatedstructure in C2. If we set G˜ν := Λν ◦Aν(D ∩ U) then by Lemma 4 we have limν→∞ G˜ν = Gand limν→∞ J˜ν = Jst, uniformly on compact subsets of C2. If F ν := Λν ◦ Aν ◦ fν thenF ν ∈ Diff(J,J˜ν)((fν)−1(D ∩ U), G˜ν) and according to Proposition 2 the sequence (F ν)νconverges after extraction to a (J, Jst)-holomorphic map F from D to G.We use the following quantitative version of Proposition 2.3.6 of [10] :8 HERVE´ GAUSSIER AND ALEXANDRE SUKHOVProposition 3. Let D′ be a domain in Cn. There is a positive constant δ0 such that forevery almost complex structure J ′ in a neighborhood of D′ satisfying ‖J ′− Jst‖C2(D′) ≤ δ0 wehave(7) ‖fJ′‖C1(K) ≤ CK,δ0‖fJ′‖C0(K),for every fJ′ ∈ O(Jst,J′)(∆, D′) and for every K ⊂⊂ ∆, where CK,δ0 is a positive constantdepending only on K and δ0.Let U ′ ⊂⊂ U be a neighborhood of p such that(8) ‖J − Jst‖C2(U ′) ≤ δ0.Fix 0 < r < 1, sufficiently close to 1. Since limν→∞ ‖J˜ν − Jst‖C2(B¯(0,r)) = 0, it follows from[7] Section 4.5a that there is a covering Rν of B¯(0, r) by J˜ν -holomorphic discs centered at theorigin for sufficiently large ν, these discs being small deformations of the straight holomorphicdiscs in the ball. More precisely there exists 0 ∈ Kr ⊂⊂ ∆ and there exists a positive constantcr such that for sufficiently large ν we have B¯(0, r) ⊂ ∪f∈Rνf(Kr) and(9) inff∈Rν ,ζ∈Kr‖df(ζ)(∂/∂x)‖ ≥ cr.Moreover we may assume that Φ−1(B¯(0, r)) ⊂ Gν . For f ∈ Rν consider the (Jst, J)-holomorphic map gν := (F ν)−1 ◦ Φ−1 ◦ f from ∆ to D. Since gν(0) = x0 it follows fromLemma 7that there exists ν0 ≥ 1 such that fν0 ◦ gν(Kr) ⊂ U ′, uniformly with respect toν >> 1. It follows now from condition (8) and from Proposition 3 (D ∩ U ′ ⊂ C2) thatthere exists a positive constant Cr such that ‖fν0 ◦ gν‖C1(Kr) ≤ Cr for sufficiently large ν, orequivalently that ‖gν‖C1(Kr) ≤ C′r with C′r > 0.It follows from inequality (9) that ‖(F ν)−1 ◦ Φ−1‖C1(B¯(0,r)) ≤ C′′r for a positive constantC′′r and finally that ‖(F ν)−1‖C1(Φ−1(B¯(0,r))) ≤ C˜r for every sufficiently large ν, where C˜r is apositive constant.Since D is relatively compact in M , Ascoli Theorem implies that some subsequence of((F ν)−1)ν converges to a (Jst, J)-holomorphic map F˜ from G to D¯. The end of the proofof Theorem 2 follows line by line the proof of Proposition 2 (iii). Since F˜ ◦ F = idD andF ◦ F˜ = idG we obtain that F ∈ Diff(J,Jst)(D,G). This gives the contradiction. References[1] Audin, M., Lafontaine, J., Holomorphic curves in symplectic geometry, Birkha¨user, Progress in Math-ematics, 117, 1994.[2] Berteloot, F., Attraction des disques analytiques et continuite´ ho¨lde´rienne d’applications holomorphespropres. (French) [Attraction of analytic disks and Holder continuity of proper holomorphic mappings]Topics in complex analysis (Warsaw, 1992), 91–98, Banach Center Publ., 31, Polish Acad. Sci., Warsaw,1995.[3] Berteloot, F., Coeure´, G., Domaines de C2 ,pseudoconvexes et de type fini ayant un groupe noncompact d’automorphismes, Ann. Inst. Fourier 41 (1991), 77-86.[4] Gaussier, H., Kim, K.T., Krantz, S.G., A note on the Wong-Rosay theorem in complex manifolds,Complex Var. Theory Appl. 47 (2002), no. 9, 761–768.[5] Gaussier, H., Sukhov, A., Estimates of the Kobayashi metric on almost complex manifolds,Pre´publication LATP 03-011.[6] Kruglikov, B.S., Existence of close pseudoholomorphic disks for almost complex manifolds and theirapplication to the Kobayashi-Royden pseudonorm. (Russian) Funktsional. Anal. i Prilozhen. 33 (1999),no. 1, 46–58, 96; translation in Funct. Anal. Appl. 33 (1999), no. 1, 38–48.[7] Nijenhuis, A., Woolf, W., Some integration problems in almost-complex and complex manifolds, Ann.Math., 77 (1963), 424-489.[8] Pinchuk, S., The scaling method and holomorphic mappings, Several complex variables and complexgeometry, Part 1 (Santa Cruz, CA, 1989), 151–161, Proc. Sympos. Pure Math., 52, Part 1, Amer. Math.Soc., Providence, RI, 1991.[9] Rosay, J.P., Sur une caracte´risation de la boule parmi les domaines de Cn par son grouped’automorphismes, Ann. Inst. Fourier, 29 (1979), 91-97.WONG-ROSAY THEOREM IN ALMOST COMPLEX MANIFOLDS 9[10] Sikorav, J.C., Some properties of holomorphic curves in almost complex manifolds, pp.165-189, inHolomorphic curves in symplectic geometry, Miche`le Audin, Jacques Lafontaine Editors, Birkha¨user(1994).[11] Wong, B., Characterization of the unit ball in Cn by its automorphism group, Invent. Math., 41 (1977),no. 3, 253-257.Herve´ Gaussier Alexandre SukhovC.M.I. U.S.T.L.39, rue Joliot-Curie, Cite´ Scientifique13453 Marseille Cedex 13 59655 Villeneuve d’Ascq Cedexgaussier@cmi.univ-mrs.fr sukhov@agat.univ-lille1.fr

Wong-Rosay Theorem in almost complex manifolds

Abstract

Similar works

Full text

Available Versions

HAL AMU