Levi-flat filling of real two-spheres in symplectic manifolds (I)
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 20 (2011) no. 3, pp. 515-539.

Soit (M,J,ω) une variété dont la structure presque complexe J est contrôlée par la forme symplectique ω. On suppose M de dimension complexe deux, Levi-convexe et à géométrie bornée. On démontre que toute 2-sphère possédant deux points elliptiques et plongée dans le bord de M est feuilletée par des bords de disques pseudoholomorphes.

Let (M,J,ω) be a manifold with an almost complex structure J tamed by a symplectic form ω. We suppose that M has the complex dimension two, is Levi-convex and with bounded geometry. We prove that a real two-sphere with two elliptic points, embedded into the boundary of M can be foliated by the boundaries of pseudoholomorphic discs.

@article{AFST_2011_6_20_3_515_0,
     author = {Gaussier, Herv\'e and Sukhov, Alexandre},
     title = {Levi-flat filling of real two-spheres in symplectic manifolds (I)},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {515--539},
     publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 20},
     number = {3},
     year = {2011},
     doi = {10.5802/afst.1316},
     mrnumber = {2894837},
     zbl = {1242.53107},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/afst.1316/}
}
Gaussier, Hervé; Sukhov, Alexandre. Levi-flat filling of real two-spheres in symplectic manifolds (I). Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 20 (2011) no. 3, pp. 515-539. doi : 10.5802/afst.1316. http://www.numdam.org/articles/10.5802/afst.1316/

[1] Barraud (J.F.), Mazzilli (E.).— Regular type of real hypersurfaces in (almost) complex manifolds, Math. Z. 248, p. 379-405 (2004). | MR 2103540 | Zbl 1082.32017

[2] Bedford (E.), Gaveau (B.).— Envelopes of holomorphy of certain 2-spheres in C 2 . Amer. J. Math. 105, p. 975-1009 (1983). | MR 708370 | Zbl 0535.32008

[3] Bedford (E.), Klingenberg (W.).— On the envelope of holomorphy of a 2-sphere in C 2 . J. Amer. Math. Soc. 4, p.  623-646 (1991). | MR 1094437 | Zbl 0736.32009

[4] Bishop (E.).— Differentiable manifolds in complex Euclidean space. Duke Math. J. 32, p. 1-21 (1965). | MR 200476 | Zbl 0154.08501

[5] Bojarski (B.V.).— Generalized solutions of a system of differential equations of the first order and elliptic type with discontinuous coefficients. Translated from the 1957 Russian original. With a foreword by Eero Saksman. Report. University of Jyväskylä Department of Mathematics and Statistics, 118. University of Jyväskylä, Jyväskylä, 2009. iv+64 pp. | MR 2488720 | Zbl 1173.35403

[6] Chirka (E.).— Introduction to the almost complex analysis, Lecture notes (2003).

[7] Diederich (K.), Fornaess (J.E.).— Pseudoconvex domains: bounded strictly plurisubharmonic exhaustion functions. Invent. Math. 39, p. 129-141 (1977). | MR 437806 | Zbl 0353.32025

[8] Diederich (K.), Sukhov (A.).— Plurisubharmonic exhaustion functions and almost complex Stein structures. Michigan Math. J. 56, p. 331-355 (2008). | MR 2492398 | Zbl 1161.32012

[9] Dubrovin (B.), Novikov (S.), Fomenko (A.).— Modern geometry. Methods and Applications. Part II. Springer-Verlag, N.Y. Inc. (1985). | MR 807945 | Zbl 0565.57001

[10] Eliashberg (Y.).— Filling by holomorphic discs and its applications. Geometry of low-dimensional manifolds, 2 (Durham, 1989), p. 45-67, London Math. Soc. Lecture Note Ser., 151, Cambridge Univ. Press, Cambridge (1990). | MR 1171908 | Zbl 0731.53036

[11] Eliashberg (Y.), Thurston (W.).— Confoliations. University Lecture Series, 13. American Mathematical Society, Providence, RI, (1998). x+66 pp. | MR 1483314 | Zbl 0893.53001

[12] Fornaess (J.E.), Ma (D.).— A 2-sphere in C 2 that cannot be filled in with analytic disks. Internat. Math. Res. Notices 1, p. 17-22 (1995). | MR 1317640 | Zbl 0866.57023

[13] Gromov (M.).— Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82, p. 307-347 (1985). | MR 809718 | Zbl 0592.53025

[14] Hind (R.).— Filling by holomorphic disks with weakly pseudoconvex boundary conditions. Geom. Funct. Anal. 7, p. 462-495 (1997). | MR 1466335 | Zbl 0884.53024

[15] Hofer (H.).— Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three. Invent. Math. 114, p. 515-563 (1993). | MR 1244912 | Zbl 0797.58023

[16] Hofer (H.), Lizan (V.), Sikorav (J.C.).— On genericity for holomorphic curves in four-dimensional almost-complex manifolds The Journal of Geometric Analysis 7, p. 149-159 (1998). | MR 1630789 | Zbl 0911.53014

[17] Kenig (C.), Webster (S.).— The local hull of holomorphy of a surface in the space of two complex variables. Invent. Math. 67, p. 1-21 (1982). | MR 664323 | Zbl 0489.32007

[18] Micallef (M.), White (B.).— The structure of branch points in minimal surfaces and in pseudoholomorphic curves. Ann. of Math. (2) 141, p. 35-85 (1995). | MR 1314031 | Zbl 0873.53038

[19] McDuff (D.).— Singularities and positivity of intersections of J-holomorphic curves. With an appendix by Gang Liu. Progr. Math., 117, Holomorphic curves in symplectic geometry, p. 191-215, Birkhäuser, Basel (1994). | MR 1274930

[20] McDuff (D.), Salamon (D.).— J-holomorphic curves and symplectic topology. American Mathematical Society Colloquium Publications, 52. American Mathematical Society, Providence, RI, 2004. xii+669 pp. | MR 2045629 | Zbl 1064.53051

[21] Sikorav (J.C.).— Some properties of holomorphic curves in almost complex manifolds. Holomorphic curves in symplectic geometry, 165-189, Progr. Math., 117, Birkhäuser, Basel (1994). | MR 1274929

[22] Sukhov (A.), Tumanov (A.).— Filling hypersurfaces by discs in almost complex manifolds of dimension 2. Indiana Univ. Math. J. 57, p. 509-544 (2008). | MR 2400266 | Zbl 1187.32021

[23] Sukhov (A.), Tumanov, (A.).— Pseudoholomorphic discs near an elliptic point. Tr. Mat. Inst. Steklova 253 (2006), Kompleks. Anal. i Prilozh., p. 296-303; translation in Proc. Steklov Inst. Math., no. 2 (253), p. 275-282 (2006). | MR 2338704

[24] Vekua (I.N.).— Generalized analytic functions. Pergamon Press, London-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass. 1962 xxix+668 pp. | MR 150320 | Zbl 0100.07603

[25] Ye (R.).— Filling by holomorphic curves in symplectic 4-manifolds. Trans. Amer. Math. Soc. 350, p. 213-250 (1998). | MR 1422913 | Zbl 0936.53047