Convergent normal form for real hypersurfaces at generic Levi degeneracy

TL;DR

This paper develops a complete convergent normal form for real hypersurfaces in complex space at generic Levi degeneracy, introducing degenerate chains as distinguished curves, marking a first in the field.

Contribution

It provides the first convergent normal form for Levi-degenerate hypersurfaces and introduces degenerate chains, advancing the understanding of hypersurface geometry.

Findings

Constructed a complete convergent normal form at generic Levi degeneracy.

Introduced the concept of degenerate chains as distinguished curves.

First normal form of its kind for Levi-degenerate hypersurfaces.

Abstract

We construct a complete convergent normal form for a real hypersurface in $\CC{N},\,N\geq 2$ at generic Levi degeneracy. This seems to be the first convergent normal form for a Levi-degenerate hypersurface. In particular, we obtain, in the spirit of the work of Chern and Moser \cite{chern}, distinguished curves in the Levi degeneracy set, that we call \it degenerate chains.