Local normal forms of singular Levi-flat hypersurfaces

TL;DR

This paper establishes new rigid normal forms for singular Levi-flat hypersurfaces defined by complex quasi-homogeneous polynomials, extending previous work and introducing volume-preserving transformations.

Contribution

It introduces two novel rigid normal forms for singular Levi-flat hypersurfaces preserved by volume-preserving coordinate changes.

Findings

Existence of rigid normal forms for hypersurfaces defined by quasi-homogeneous polynomials.

Generalization of previous results by Burns-Gong and Fernández-Pérez.

Introduction of volume-preserving normal forms.

Abstract

We study normal forms of germs of singular real-analytic Levi-flat hypersurfaces. We prove the existence of rigid normal forms for singular Levi-flat hypersurfaces which are defined by the vanishing of the real part of complex quasi-homogeneous polynomials with isolated singularity. This result generalizes previous results of Burns-Gong and Fern\'andez-P\'erez. Furthermore, we prove the existence of two new rigid normal forms for singular real-analytic Levi-flat hypersurfaces which are preserved by a change of isochore coordinates, that is, a change of coordinates that preserves volume.