Gluing complex discs to Lagrangian manifolds by Gromov’s method
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 22 (2013) no. 4, p. 811-842

The paper discusses some aspects of Gromov’s theory of gluing complex discs to Lagrangian manifolds.

L’article discute certains aspects de la théorie d’attachement des disques complexes aux variétés Lagrangiennes par la méthode de Gromov.

@article{AFST_2013_6_22_4_811_0,
     author = {Sukhov, Alexandre and Tumanov, Alexander},
     title = {Gluing complex discs to Lagrangian manifolds by Gromov's method},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 22},
     number = {4},
     year = {2013},
     pages = {811-842},
     doi = {10.5802/afst.1389},
     mrnumber = {3137251},
     zbl = {1287.32011},
     language = {en},
     url = {http://www.numdam.org/item/AFST_2013_6_22_4_811_0}
}
Sukhov, Alexandre; Tumanov, Alexander. Gluing complex discs to Lagrangian manifolds by Gromov’s method. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 22 (2013) no. 4, pp. 811-842. doi : 10.5802/afst.1389. http://www.numdam.org/item/AFST_2013_6_22_4_811_0/

[1] Alexander (H.).— Gromov’s method and Bennequin’s problem, Invent. Math. 125, p. 135-148, (1996). | MR 1389963 | Zbl 0853.32003

[2] Arnold (V.I.).— Symplectic geometry and topology, J. Math. Phys. 41, p. 3307-3343, (2000). | MR 1768639 | Zbl 1021.53053

[3] Audin (M.), Lafontaine (J.) (Eds.).— “Holomorphic curves in symplectic geometry", Birkhauser, Progress in Mathematics, V.117 (1994). | MR 1274923 | Zbl 0802.53001

[4] Bers (L.), Schechter (M.).— “Elliptic equations", 1964 Partial differential equations, p. 131-299, Interscience, New York, (1964). | MR 165224 | Zbl 0128.09404

[5] Chirka (E.M.).— “Complex analytic sets", Kluwer, (1989). | MR 1111477 | Zbl 0683.32002

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

[7] Donaldson (S.).— What is ... a pseudoholomorphic curve?, Notices of the AMS, 52, p. 1026-1027, (2005). | MR 2168014 | Zbl 1090.32015

[8] Duval (J.) and Gayet (D.).— Riemann surfaces and totally real tori, arXiv 0910.2139.

[9] Eliashberg (Y.), Traynor (L.) (Eds.).— “Symplectic geometry and topology", AMS, Princeton, NJ. (1997). | MR 1702940

[10] Gaussier (H.), Sukhov (A.).— Levi-flat filling of real two-spheres in symplectic manifolds (I),(II), Ann. Fac. Sci. Toulouse Math. , XX(2011), p. 515-539, XXI, p. 783-816, (2012). | Numdam | MR 2894837 | Zbl 1242.53107

[11] Ivashkovich (S.), Shevchishin (V.).— “Complex curves in almost-complex manifolds and meromorphic hulls", Institut fur Math. Ruhr-Universitat Bochum, (1999).

[12] Ivashkovich (S.), Shevchishin (V.).— Reflection principle and J-complex curves with boundary on totally real immersions, Commun. Contem. Math. 4, p. 65–106, (2002). | MR 1890078 | Zbl 1025.32024

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

[14] Hofer (H.), Lizan (V.), Sikorav (J.-C.).— On genericity for holomorphic curves in four-dimensional almost-complex manifolds, J.Geom. Anal. 7, p. 144-159, (1997). | MR 1630789 | Zbl 0911.53014

[15] Hummel (Ch.).— “Gromov’s compactness theorem for pseudoholomorphic curves", Birkhauser, Progress in Mathematics V.151 (1997). | MR 1451624 | Zbl 0870.53002

[16] McDuff (D.), Salomon (D.).— “Introduction to symplectic topology", 2nd edition, Ofxord University Press, (1998). | MR 1702941 | Zbl 0844.58029

[17] McDuff (D.), Salomon (D.).— “J-holomorphic curves and symplectic topology", AMS, Coll. Publ. Vol. 52, (2004). | Zbl 1064.53051

[18] Mikallef (M.J.), White (B.).— The structure of branch points in minimal surfaces and in pseudoholomorphic curves, Ann. Math. 139, p. 35-85, (1994). | MR 1314031 | Zbl 0873.53038

[19] Mikhlin (S.), Prössdorf (S.).— “Singular integral equations", Springer Verlag, Berlin, (1986).

[20] Morrey (Ch.B.), Jr.— “Multiple integrals in the calculus of variations", Springer Verlag. N.Y. (1966). | MR 202511 | Zbl 1213.49002

[21] Smale (S.).— An infinite dimensinal version of Sard’s theorem, Amer. J. Math. 87, p. 861-866, (1965). | MR 185604 | Zbl 0143.35301

[22] Sukhov (A.), Tumanov (A.).— Regularization of almost complex structures, Ann. Sc. Norm. Sup. Pisa. 10, p. 389-411, (2011). | MR 2856153 | Zbl 1228.32016

[23] Sukhov (A.), Tumanov (A.).— Deformation and transversality of pseudo-holomorphic discs , J. Anal. Math. 116, p. 1-16, (2012). | MR 2892615 | Zbl 1273.32032

[24] Sukhov (A.), Tumanov (A.).— Boundary value problems and J-complex curves, arXiv 1103.0324, to appear in Complex Variables and Elliptic Equations (Special Issue). | Zbl pre06231132

[25] Vekua (I. N.).— “Generalized analytic functions", Pergamon Press, (1962). | MR 150320 | Zbl 0100.07603

[26] Wendland (W.L.).— “Elliptic systems in the plane", Pitman, (1979). | MR 518816 | Zbl 0396.35001