Levi-flat filling of real two-spheres in symplectic manifolds (II)
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 21 (2012) no. 4, pp. 783-816.

On considère une variété presque complexe (M,J,ω) avec la frontière Levi convexe M et une tame forme symplectique ω. Soit S 2 une 2-sphere réelle avec des points elliptiques et hyperboliques, plongée génériquement dans M. On démontre un résultat sur le remplissage de S 2 par des disques holomorphes.

We consider a compact almost complex manifold (M,J,ω) with smooth Levi convex boundary M and a symplectic tame form ω. Suppose that S 2 is a real two-sphere, containing complex elliptic and hyperbolic points and generically embedded into M. We prove a result on filling S 2 by holomorphic discs.

@article{AFST_2012_6_21_4_783_0,
     author = {Gaussier, Herv\'e and Sukhov, Alexandre},
     title = {Levi-flat filling of real two-spheres in symplectic manifolds {(II)}},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {783--816},
     publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 21},
     number = {4},
     year = {2012},
     doi = {10.5802/afst.1351},
     mrnumber = {3052031},
     zbl = {1260.53138},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/afst.1351/}
}
TY  - JOUR
AU  - Gaussier, Hervé
AU  - Sukhov, Alexandre
TI  - Levi-flat filling of real two-spheres in symplectic manifolds (II)
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2012
DA  - 2012///
SP  - 783
EP  - 816
VL  - Ser. 6, 21
IS  - 4
PB  - Université Paul Sabatier, Institut de mathématiques
PP  - Toulouse
UR  - http://archive.numdam.org/articles/10.5802/afst.1351/
UR  - https://www.ams.org/mathscinet-getitem?mr=3052031
UR  - https://zbmath.org/?q=an%3A1260.53138
UR  - https://doi.org/10.5802/afst.1351
DO  - 10.5802/afst.1351
LA  - en
ID  - AFST_2012_6_21_4_783_0
ER  - 
Gaussier, Hervé; Sukhov, Alexandre. Levi-flat filling of real two-spheres in symplectic manifolds (II). Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 21 (2012) no. 4, pp. 783-816. doi : 10.5802/afst.1351. http://archive.numdam.org/articles/10.5802/afst.1351/

[1] Arnold (V.I.).— On a characteristic class appearing in the quantization condition, Funktsion. Anal. i Prilozhen. 1, p. 1-14 (1967). | MR 211415 | Zbl 0175.20303

[2] 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

[3] Chirka (E.).— Lectures on almost complex analysis, Lecture notes (2003).

[4] Chirka (E.).— Complex analytic sets. Kluwer (1989). | MR 1111477 | Zbl 0683.32002

[5] 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

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

[7] 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

[8] Forstneric (F.).— Complex tangents of real surfaces in complex surfaces. Duke Math. J. 67, p. 353-376 (1992). | MR 1177310 | Zbl 0761.53032

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

[10] Gaussier (H.), Sukhov (A.).— Levi-flat filling of real two-spheres in symplectic manifolds (I). Annales Fac. Sci. Toulouse, XX, p. 515-539 (2011). | Numdam | MR 2894837 | Zbl 1242.53107

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

[12] 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

[13] Ivashkovich (S.), Shevchishin (V.).— Gromov’s compactness theorem for J-complex curves with boundary, IMRN, 22 (2000). | Zbl 0994.53010

[14] Kruzhilin (N.).— Two-dimensional spheres in the boundaries of strictly pseudoconvex domains in 2 . Math. USSR Izvetsia 39, 1151-1187 (1992). | MR 1152210 | Zbl 0778.32003

[15] Labourie (F.).— Exemples de courbes pseudo-holomorphes en géométrie riemannienne, in Holomorphic curves in symplectic geometry, p. 251-269, Progr. Math. 117, Birkhäuser, Basel (1994). | MR 1274933

[16] 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

[17] Nemirovski (S.).— Complex analysis and differential topology on complex surfaces. Russian Math. Surveys 54, 729-752 (1999). | MR 1741278 | Zbl 0971.32016

[18] Pansu (P.).— Compactness. in Holomorphic curves in symplectic geometry, 233-249, Progr. Math., 117, Birkhäuser, Basel (1994). | MR 1274932

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

[20] Sukhov (A.), Tumanov (A.).— Regularization of almost complex structures and gluing holomorphic discs to tori,Ann. Scuola Norm. Sup. Pisa (5),X, p. 389-411 (2011). | MR 2856153 | Zbl 1228.32016

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

Cité par Sources :