A Morse-Bott normal form for real analytic Levi-flat hypersurfaces

TL;DR

This paper establishes a normal form for real-analytic Levi-flat hypersurfaces with Morse-Bott singularities, extending existing results and introducing a new normal form for specific quadratic cases.

Contribution

It provides a Morse-Bott normal form for Levi-flat hypersurfaces, generalizing previous results and introducing a novel normal form for quadratic cases.

Findings

Recovered Burns-Gong normal form for generic Morse singularities

Established a normal form for hypersurfaces with Morse-Bott singularities

Introduced a new normal form for certain quadratic Levi-flat hypersurfaces

Abstract

We prove the existence of a normal form for a real-analytic Levi-flat hypersurface defined by the vanishing of the real part of a holomorphic function with a Morse-Bott singularity. As a consequence, we recover the Burns-Gong normal form for Levi-flat hypersurfaces with generic Morse singularities and provide a new normal form for a certain class of real analytic quadratic Levi-flat hypersurfaces.