Regularization of almost complex structures and gluing holomorphic discs to tori
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 2, p. 389-411

We prove a result on removing singularities of almost complex structures pulled back by a non-diffeomorphic map. As an application we prove the existence of global J-holomorphic discs with boundaries attached to real tori.

Published online : 2018-08-07
Classification:  32H02,  53C15
@article{ASNSP_2011_5_10_2_389_0,
     author = {Sukhov, Alexandre and Tumanov, Alexander},
     title = {Regularization of almost complex structures and gluing holomorphic discs to tori},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 10},
     number = {2},
     year = {2011},
     pages = {389-411},
     zbl = {1228.32016},
     mrnumber = {2856153},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2011_5_10_2_389_0}
}
Sukhov, Alexandre; Tumanov, Alexander. Regularization of almost complex structures and gluing holomorphic discs to tori. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 2, pp. 389-411. http://www.numdam.org/item/ASNSP_2011_5_10_2_389_0/

[1] H. Alexander, Discs with boundaries in totally real and Lagrangian manifolds, Duke Math. J. 100 (1999), 131–138. | MR 1714757 | Zbl 0953.32026

[2] M. Cerne, Non-linear Riemann-Hilbert problem for bordered Riemann surfaces, Amer. J. Math. 126 (2004), 65–87. | MR 2033564 | Zbl 1062.30044

[3] B. Coupet, A. Sukhov and A. Tumanov, Proper J-holomorphic discs in Stein domains of dimension 2, Amer. J. Math. 131 (2009), 653–674. | MR 2530850 | Zbl 1168.32020

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

[5] B. Drinovec Drnovšek and F. Forstnerič, Holomorphic curves in complex spaces, Duke Math. J. 139 (2007), 203–253. | MR 2352132 | Zbl 1133.32002

[6] J. Duval, Un théorème de Green presque complexe, Ann. Inst. Fourier (Grenoble) 54 (2004), 2357–2367. | Numdam | MR 2139696 | Zbl 1076.32020

[7] J. Duval, personal communication.

[8] F. Forstnerič, Polynomial hulls of sets fibered over the circle, Indiana Univ. Math. J. 37 (1988), 869–889. | MR 982834 | Zbl 0647.32017

[9] F. Forstnerič and J. Globevnik, Discs in pseudoconvex domains, Comment. Math. Helv. 67 (1992), 129–145. | MR 1144617 | Zbl 0779.32016

[10] M. Gromov, Pseudo-holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307–347. | MR 809718 | Zbl 0592.53025

[11] S. Ivashkovich and J.-P. Rosay, Schwarz-type lemmas for solutions of ¯-inequalities and complete hyperbolicity of almost complex structures, Ann. Inst. Fourier (Grenoble) 54 (2004), 2387–2435. | Numdam | MR 2139698 | Zbl 1072.32007

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

[13] S. Ivashkovich and A. Sukhov, Schwarz reflection principle and boundary uniqueness for J-complex curves, Ann. Inst. Fourier (Grenoble) 60 (2010), 1489–1513. | Numdam | MR 2722249 | Zbl 1208.32026

[14] L. Lempert, Analytic continuation in mapping spaces, Pure Appl. Math. Q. 6 (2010), 1051–1080. | MR 2742039 | Zbl 1222.32022

[15] V. N. Monakhov, “Boundary Problems with Free Boundary for Elliptic Systems of Equations”, Translations of Mathematical Monographs, Vol. 57, Amer. Math. Soc., Providence, 1983. | MR 717387 | Zbl 0532.35001

[16] A. Nijenhuis and W. Woolf, Some integration problems in almost-complex and complex manifolds, Ann. of Math. (2) 77 (1963), 429–484. | MR 149505 | Zbl 0115.16103

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

[18] I. N. Vekua, “Generalized Analytic Functions”, Pergamon, 1962. | MR 150320 | Zbl 0100.07603