J-holomorphic discs and real analytic hypersurfaces
Annales de l'Institut Fourier, Volume 64 (2014) no. 5, pp. 2223-2250.

We give in 6 a real analytic almost complex structure J, a real analytic hypersurface M and a vector v in the Levi null set at 0 of M, such that there is no germ of J-holomorphic disc γ included in M with γ(0)=0 and γ x(0)=v, although the Levi form of M has constant rank. Then for any hypersurface M and any complex structure J, we give sufficient conditions under which there exists such a germ of disc.

Nous définissons dans 6 une structure presque complexe réelle analytique J, une hypersurface réelle analytique M dont la forme de Levi est de rang constant et un vecteur v appartenant au noyau de la forme de Levi de M en 0 tels qu’il n’existe pas de germe de disque J-holomorphe γ inclus dans M vérifiant γ(0)=0 et γ x(0)=v. Nous donnons ensuite des conditions suffisantes pour qu’un tel germe de disque existe.

DOI: 10.5802/aif.2910
Classification: 32Q60, 32Q65
Keywords: almost complex structure, $J$-holomorphic disc, hypersurface
Mot clés : Structure presque complexe, disque $J$-holomorphe, hypersurface

Alexandre, William 1; Mazzilli, Emmanuel 1

1 Laboratoire Paul Painlevé U.M.R. CNRS 8524 U.F.R. de Mathématiques cité scientifique Université Lille 1 F59 655 Villeneuve d’Ascq Cedex, France.
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Alexandre, William; Mazzilli, Emmanuel. $J$-holomorphic discs and real analytic hypersurfaces. Annales de l'Institut Fourier, Volume 64 (2014) no. 5, pp. 2223-2250. doi : 10.5802/aif.2910. https://aif.centre-mersenne.org/articles/10.5802/aif.2910/

[1] Barraud, J.-F.; Mazzilli, E. Regular type of real hyper-surface in (almost) complex manifolds, Math. Z., Volume 248 (2004), pp. 757-772 | DOI | MR | Zbl

[2] Freeman, M. Local Complex foliation of Real Submanifolds, Math. Ann., Volume 209 (1974), pp. 1-30 | DOI | MR | Zbl

[3] Ivashkovich, S.; Rosay, J.-P. Schwarz-type lemmas for solutions of ¯-inequalities and complete hyperbolicity of almost complex manifolds, Ann. Inst. Fourier (Grenoble), Volume 54 (2004) no. 7, pp. 2387-2435 (2005) | DOI | Numdam | MR | Zbl

[4] Ivashkovich, S.; Sukhov, A. Schwarz reflection principle, boundary regularity and compactness for J-complex curves, Ann. Inst. Fourier (Grenoble), Volume 60 (2010) no. 4, pp. 1489-1513 | DOI | Numdam | MR | Zbl

[5] Kruzhilin, N.; Sukhov, A. Pseudoholomorphic discs attached to CR-submanifolds of almost complex spaces, Bull. Sci. Math., Volume 129 (2005) no. 5, pp. 398-414 | DOI | MR | Zbl

[6] Loomis, L. H. An introduction to abstract harmonic analysis, D. Van Nostrand Company, Inc., Toronto-New York-London, 1953 | MR | Zbl

[7] Nagano, T. Linear differential systems with singularities and an application to transitive Lie algebras, J. Math. Soc. Japan, Volume 18 (1966), pp. 398-404 | DOI | MR | Zbl

[8] Sikorav, J.-C. Some properties of holomorphic curves in almost complex manifolds, Holomorphic curves in symplectic geometry (Prog. Math.), Volume 117, Birkhauser, Basel, 1994, pp. 165-189 | MR

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