$J$-holomorphic discs and real analytic hypersurfaces

Alexandre, William; Mazzilli, Emmanuel

doi:10.5802/aif.2910

J

-holomorphic discs and real analytic hypersurfaces
[Disques

J

-holomorphes et hypersurfaces rÃ©elles analytiques]

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.

Annales de l'Institut Fourier, Tome 64 (2014) no. 5, pp. 2223-2250.

Résumé
Abstract

Nous définissons dans $ℝ^{6}$ 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 $\frac{\partial γ}{\partial x} (0) = v$ . Nous donnons ensuite des conditions suffisantes pour qu’un tel germe de disque existe.

We give in $ℝ^{6}$ 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 $\frac{\partial γ}{\partial x} (0) = v$ , 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.

MR Zbl

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

Affiliations des auteurs :

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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     title = {$J$-holomorphic discs and real analytic hypersurfaces},
     journal = {Annales de l'Institut Fourier},
     pages = {2223--2250},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {64},
     number = {5},
     year = {2014},
     doi = {10.5802/aif.2910},
     zbl = {06387337},
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Alexandre, William; Mazzilli, Emmanuel. $J$-holomorphic discs and real analytic hypersurfaces. Annales de l'Institut Fourier, Tome 64 (2014) no. 5, pp. 2223-2250. doi : 10.5802/aif.2910. http://www.numdam.org/articles/10.5802/aif.2910/

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