Disques J-holomorphes contenus dans une hypersurface

TL;DR

This paper investigates the presence of germs of J-holomorphic curves within real analytic hypersurfaces in 4-dimensional symplectic manifolds, establishing conditions under which such germs do not exist.

Contribution

It proves that under certain topological conditions, compact hypersurfaces in symplectic manifolds are of finite type, preventing germs of J-holomorphic curves from existing.

Findings

Compact hypersurfaces are of finite type under topological hypotheses.

No germs of J-holomorphic curves exist on such hypersurfaces.

Generalizes classical results from complex analysis to symplectic geometry.

Abstract

We study germs of J-Holomorphic curves contained in $M$, a real analytic hypersurface of an symplectic manifold of dimension 4. We show, under topological hypothesis on $M$, that if $M$ is compact then $M$ is of finite type and so there is no germs of $J$-holomorphic curves on $M$(with $J$ adapted with the symplectic form). In $\Bbb{C}^2$ with the standard complex structure, this is a classical result of Diederich-Fornaess.