We give a necessary condition for the existence of extremal pseudoholomorphic discs on some domains in where is a small almost complex deformation of the standard complex structure.
@article{ASNSP_2010_5_9_4_759_0,
author = {Gaussier, Herv\'e and Joo, Jae-Cheon},
title = {Extremal discs in almost complex spaces},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {759--783},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 9},
number = {4},
year = {2010},
zbl = {1209.32003},
mrnumber = {2789474},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2010_5_9_4_759_0/}
}
Gaussier, Hervé; Joo, Jae-Cheon. Extremal discs in almost complex spaces. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 4, pp. 759-783. http://www.numdam.org/item/ASNSP_2010_5_9_4_759_0/
[1] , A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl. (9) 36 (1957), 235–249. | MR 92067 | Zbl 0084.30402
[2] and , “Holomorphic Curves in Symplectic Geometry”, Progr. Math., Vol. 117, Birkhäuser, Basel, 1994. | MR 1274923 | Zbl 0802.53001
[3] and , Moduli for pointed convex domains, Invent. Math. 104 (1991), 61–112. | EuDML 143875 | MR 1094047 | Zbl 0731.32010
[4] and , Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary, J. Amer. Math. Soc. 7 (1994), 661–676. | MR 1242454 | Zbl 0807.32008
[5] , The Kobayashi geometry of Lee’s model of dimension 6, preprint.
[6] , and , Fefferman’s mapping theorem on almost complex manifolds in complex dimension two, Math. Z. 250 (2005), 59–90. | MR 2136668 | Zbl 1076.32028
[7] , and , Riemann maps in almost complex manifolds, Ann. Scuola Norm. Super. Pisa Cl. Sci. (5) 2 (2003), 761–785. | EuDML 84518 | Numdam | MR 2040642 | Zbl 1170.32310
[8] and , On the geometry of model almost complex manifolds with boundary, Math. Z. 254 (2006), 567–589. | MR 2244367 | Zbl 1107.32009
[9] , Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307–347. | EuDML 143289 | MR 809718 | Zbl 0592.53025
[10] and , On the local behavior of solutions of non-parabolic partial differential equations, Amer. J. Math. 75 (1953), 449–476. | MR 58082 | Zbl 0052.32201
[11] and , “Tangent and Cotangent Bundles: Differential Geometry”, Pure and Applied Mathematics, No. 16, Marcel Dekker, Inc., New York, 1973. | MR 350650 | Zbl 0262.53024
[12] and , Boundary Values and Boundary Uniqueness of -Holomorphic Mappings, arXiv:0902.4800. | MR 2836395 | Zbl 1227.32032
[13] , Strongly pseudoconvex homogeneous domains in almost complex manifolds, J. Reine Angew. Math. 623 (2008), 123–160. | MR 2458042 | Zbl 1160.32024
[14] , La métrique de Kobayashi et la représentation des domaines sur la boule, (French) [The Kobayashi metric and the representation of domains on the ball] Bull. Soc. Math. France 109 (1981), 427–474. | EuDML 87405 | Numdam | MR 660145 | Zbl 0492.32025
[15] , A precise result on the boundary regularity of biholomorphic mappings, Math. Z. 193 (1986), 559–579. | EuDML 173805 | MR 867348 | Zbl 0603.32013
[16] , Holomorphic invariants, normal forms, and the moduli space of convex domains, Ann. of Math. 128 (1988), 43–78. | MR 951507 | Zbl 0658.32015
[17] and , The tangent bundle of an almost complex manifold, Canad. Math. Bull. 44 (2001), 70–79. | MR 1816050 | Zbl 0984.53029
[18] and , “-holomorphic Curves and Symplectic Topology”, Amer. Math. Soc., Coll. Publ., Vol. 52, 2004. | MR 2045629 | Zbl 1064.53051
[19] M. -Y. Pang, Smoothness of the Kobayashi metric of nonconvex domains, Internat. J. Math. 4 (1993), 953–987. | MR 1250257 | Zbl 1093.74062
[20] , Real and complex Monge Ampère equations and the geometry of strictly convex domains, Publ. Res. Inst. Math. Sci. 20 (1984), 867–875. | MR 762955 | Zbl 0578.32002
[21] , The Euler Lagrange equations for extremal holomorphic mappings of the unit disk, Michigan Math. J. 30 (1983), 317–333. | MR 725784
[22] , Total reality of conormal bundles of hypersurfaces in almost complex manifolds, Int. J. Geom. Methods Mod. Phys. 3 (2006), 1255–1262. | MR 2264414 | Zbl 1112.53057
[23] and , An existence theorem for stationary discs in almost complex manifolds, J. Math. Anal. Appl. 327 (2007), 269–286. | MR 2277411 | Zbl 1126.32021
[24] , Extremal discs and the regularity of CR mappings in higher codimension, Amer. J. Math. 123 (2001), 445–473. | MR 1833148 | Zbl 0995.32024
[25] , “Generalized Analytic Functions”, Pergamon Press, London-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass. 1962. | MR 150320
