Soit une variété dont la structure presque complexe est contrôlée par la forme symplectique . On suppose 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 est feuilletée par des bords de disques pseudoholomorphes.
Let be a manifold with an almost complex structure tamed by a symplectic form . We suppose that 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 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}, zbl = {1242.53107}, mrnumber = {2894837}, language = {en}, url = {http://www.numdam.org/articles/10.5802/afst.1316/} }
TY - JOUR AU - Gaussier, Hervé AU - Sukhov, Alexandre TI - Levi-flat filling of real two-spheres in symplectic manifolds (I) JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2011 SP - 515 EP - 539 VL - 20 IS - 3 PB - Université Paul Sabatier, Institut de mathématiques PP - Toulouse UR - http://www.numdam.org/articles/10.5802/afst.1316/ DO - 10.5802/afst.1316 LA - en ID - AFST_2011_6_20_3_515_0 ER -
%0 Journal Article %A Gaussier, Hervé %A Sukhov, Alexandre %T Levi-flat filling of real two-spheres in symplectic manifolds (I) %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2011 %P 515-539 %V 20 %N 3 %I Université Paul Sabatier, Institut de mathématiques %C Toulouse %U http://www.numdam.org/articles/10.5802/afst.1316/ %R 10.5802/afst.1316 %G en %F AFST_2011_6_20_3_515_0
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 | Zbl
[2] Bedford (E.), Gaveau (B.).— Envelopes of holomorphy of certain -spheres in . Amer. J. Math. 105, p. 975-1009 (1983). | MR | Zbl
[3] Bedford (E.), Klingenberg (W.).— On the envelope of holomorphy of a -sphere in . J. Amer. Math. Soc. 4, p. 623-646 (1991). | MR | Zbl
[4] Bishop (E.).— Differentiable manifolds in complex Euclidean space. Duke Math. J. 32, p. 1-21 (1965). | MR | Zbl
[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 | Zbl
[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 | Zbl
[8] Diederich (K.), Sukhov (A.).— Plurisubharmonic exhaustion functions and almost complex Stein structures. Michigan Math. J. 56, p. 331-355 (2008). | MR | Zbl
[9] Dubrovin (B.), Novikov (S.), Fomenko (A.).— Modern geometry. Methods and Applications. Part II. Springer-Verlag, N.Y. Inc. (1985). | MR | Zbl
[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 | Zbl
[11] Eliashberg (Y.), Thurston (W.).— Confoliations. University Lecture Series, 13. American Mathematical Society, Providence, RI, (1998). x+66 pp. | MR | Zbl
[12] Fornaess (J.E.), Ma (D.).— A -sphere in that cannot be filled in with analytic disks. Internat. Math. Res. Notices 1, p. 17-22 (1995). | MR | Zbl
[13] Gromov (M.).— Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82, p. 307-347 (1985). | MR | Zbl
[14] Hind (R.).— Filling by holomorphic disks with weakly pseudoconvex boundary conditions. Geom. Funct. Anal. 7, p. 462-495 (1997). | MR | Zbl
[15] Hofer (H.).— Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three. Invent. Math. 114, p. 515-563 (1993). | MR | Zbl
[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 | Zbl
[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 | Zbl
[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 | Zbl
[19] McDuff (D.).— Singularities and positivity of intersections of -holomorphic curves. With an appendix by Gang Liu. Progr. Math., 117, Holomorphic curves in symplectic geometry, p. 191-215, Birkhäuser, Basel (1994). | MR
[20] McDuff (D.), Salamon (D.).— -holomorphic curves and symplectic topology. American Mathematical Society Colloquium Publications, 52. American Mathematical Society, Providence, RI, 2004. xii+669 pp. | MR | Zbl
[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
[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 | Zbl
[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
[24] Vekua (I.N.).— Generalized analytic functions. Pergamon Press, London-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass. 1962 xxix+668 pp. | MR | Zbl
[25] Ye (R.).— Filling by holomorphic curves in symplectic -manifolds. Trans. Amer. Math. Soc. 350, p. 213-250 (1998). | MR | Zbl
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