Riemann maps in almost complex manifolds
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 4, pp. 761-785.

We prove the existence of stationary discs in the ball for small almost complex deformations of the standard structure. We define a local analogue of the Riemann map and establish its main properties. These constructions are applied to study the local geometry of almost complex manifolds and their morphisms.

Classification: 32H02, 32H40, 32T15, 53C15, 53D12
@article{ASNSP_2003_5_2_4_761_0,
     author = {Coupet, Bernard and Gaussier, Herv\'e and Sukhov, Alexandre},
     title = {Riemann maps in almost complex manifolds},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {761--785},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 2},
     number = {4},
     year = {2003},
     mrnumber = {2040642},
     zbl = {1170.32310},
     language = {en},
     url = {http://archive.numdam.org/item/ASNSP_2003_5_2_4_761_0/}
}
TY  - JOUR
AU  - Coupet, Bernard
AU  - Gaussier, Hervé
AU  - Sukhov, Alexandre
TI  - Riemann maps in almost complex manifolds
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2003
SP  - 761
EP  - 785
VL  - 2
IS  - 4
PB  - Scuola normale superiore
UR  - http://archive.numdam.org/item/ASNSP_2003_5_2_4_761_0/
LA  - en
ID  - ASNSP_2003_5_2_4_761_0
ER  - 
%0 Journal Article
%A Coupet, Bernard
%A Gaussier, Hervé
%A Sukhov, Alexandre
%T Riemann maps in almost complex manifolds
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2003
%P 761-785
%V 2
%N 4
%I Scuola normale superiore
%U http://archive.numdam.org/item/ASNSP_2003_5_2_4_761_0/
%G en
%F ASNSP_2003_5_2_4_761_0
Coupet, Bernard; Gaussier, Hervé; Sukhov, Alexandre. Riemann maps in almost complex manifolds. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 4, pp. 761-785. http://archive.numdam.org/item/ASNSP_2003_5_2_4_761_0/

[1] Z. Balogh - Ch. Leuenberger, Higher dimensional Riemann maps, Internat. J. Math. 9 (1998), 421-442. | MR | Zbl

[2] D. Bennequin, Topologie symplectique, convexité holomorphe holomorphe et structures de contact [d'après Y. Eliashberg, D. Mc Duff et al.], Astérisque 189-190 (1990), 285-323. | Numdam | MR | Zbl

[3] J. Bland, Contact geometry and CR structures on 𝕊 3 , Acta Math. 172 (1994), 1-49. | MR | Zbl

[4] J. Bland - T. Duchamp, Moduli for pointed convex domains, Invent. Math. 104 (1991), 61-112. | MR | Zbl

[5] J. Bland - T. Duchamp - M. Kalka, A characterization of n by its automorphism group, Lecture Notes in Math. 1268 (1987), 60-65. | MR | Zbl

[6] M. Cerne, Stationary discs of fibrations over the circle, Internat. J. Math. 6 (1995), 805-823. | MR | Zbl

[7] E. Chirka, Regularity of boundaries of analytic sets, Math. USSR-Sb. 45 (1983), 291-336. | MR | Zbl

[8] K. Clancey - I. Gohberg, “Factorization of matrix functions and singular integral operators", Birkhauser, Basel, Boston, Stuttgart, 1981. | MR | Zbl

[9] C. Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. math. 26 (1974), 1-65. | MR | Zbl

[10] J. Globevnik, Perturbation by analytic discs along maximal real submanifolds of N , Math. Z. 217 (1994), 287-316. | MR | Zbl

[11] J. Globevnik, Perturbing analytic discs attached to a maximal totally real submanifolds of n , Indag. Math. 7 (1996), 37-46. | MR | Zbl

[12] S. Ishihara - K. Yano, “Tangent and cotangent bundles: differential geometry", Pure and Applied Mathematics, No. 16, Marcel Dekker Inc., New York, 1973. | MR | Zbl

[13] L. Lempert, La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France 109 (1981), 427-474. | Numdam | MR | Zbl

[14] L. Lempert, Solving the degenerate complex Monge-Ampère equation with one concentrated singularity, Math. Ann. 263 (1983), 515-532. | MR | Zbl

[15] L. Lempert, A precise result on the boundary regularity of biholomorphic mappings, Math. Z. 193 (1986), 559-579. | MR | Zbl

[16] L. Lempert, Holomorphic invariants, normal forms and moduli space of convex domains, Ann. of Math. 128 (1988), 47-78. | MR | Zbl

[17] L. Lempert, Erratum: A precise result on the boundary regularity of biholomorphic mappings, Math. Z. 206 (1991), 501-504. | MR | Zbl

[18] P. Libermann, Problèmes d'équivalence relatifs à une structure presque complexe sur une variété à quatre dimensions, Acad. Roy. Belgique Bull. Cl. Sci. (5) 36 (1950), 742-755. | MR | Zbl

[19] M. Y. Pang, Smoothness of the Kobayashi metric of non-convex domains, Internat. J. Math. 4 (1993), 953-987. | MR | Zbl

[20] S. Semmes, A generalization of Riemann mappings and geometric structures on a space of domains in n , Mem. Amer. Math. Soc. 98 (1992), vi+98pp. | MR | Zbl

[21] J. C. Sikorav, “Some properties of holomorphic curves in almost complex manifolds”, pp.165-189, In: “Holomorphic curves in symplectic geometry", Michèle Audin, Jacques Lafontaine Editors, Birkhäuser, 1994. | MR

[22] A. Spiro - S. Trapani, Eversive maps of bounded convex domains in n+1 , J. Geom. Anal. 12 (2002), 695-715. | MR | Zbl

[23] A. Tumanov, Extremal discs and the regularity of CR mappings in higher codimension, Amer. J. Math. 123 (2001), 445-473. | MR | Zbl

[24] N. P. Vekua, “Systems of singular integral equations", Nordholf, Groningen, 1967. | MR | Zbl

[25] S. Webster, On the reflection principle in several complex variables, Proc. Amer. Math. Soc. 71 (1978), 26-28. | MR | Zbl