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

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 $\gamma$ included in $M$ with $\gamma \left(0\right)=0$ and $\frac{\partial \gamma }{\partial x}\left(0\right)=v$, 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 ${ℝ}^{6}$ 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 $\gamma$ inclus dans $M$ vérifiant $\gamma \left(0\right)=0$ et $\frac{\partial \gamma }{\partial x}\left(0\right)=v$. Nous donnons ensuite des conditions suffisantes pour qu’un tel germe de disque existe.

DOI : https://doi.org/10.5802/aif.2910
Classification:  32Q60,  32Q65
Keywords: almost complex structure, $J$-holomorphic disc, hypersurface
@article{AIF_2014__64_5_2223_0,
author = {Alexandre, William and Mazzilli, Emmanuel},
title = {$J$-holomorphic discs and real analytic hypersurfaces},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {64},
number = {5},
year = {2014},
pages = {2223-2250},
doi = {10.5802/aif.2910},
mrnumber = {3330937},
zbl = {06387337},
language = {en},
url = {http://www.numdam.org/item/AIF_2014__64_5_2223_0}
}

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. http://www.numdam.org/item/AIF_2014__64_5_2223_0/

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