J-holomorphic discs and real analytic hypersurfaces

TL;DR

The paper constructs a specific example in 6 demonstrating the non-existence of certain J-holomorphic discs in real analytic hypersurfaces, and establishes necessary conditions for their existence.

Contribution

It provides a counterexample in 6 showing the failure of expected holomorphic disc existence and derives necessary conditions for such discs in general settings.

Findings

Counterexample in 6 with no J-holomorphic disc despite constant Levi form

Necessary conditions for the existence of J-holomorphic discs in hypersurfaces

Insights into the relationship between Levi form rank and holomorphic discs

Abstract

We give in \mathbb{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 f included in M with f(0)=0 and \frac{\partial f}{\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 necessary conditions under which there exists such a germ of disc.