New normal forms for Levi-nondegenerate hypersurfaces

TL;DR

This paper introduces a broad family of new normal forms for Levi-nondegenerate hypersurfaces in complex spaces, expanding on the classical Chern-Moser form with simpler alternatives and diverse normalization conditions.

Contribution

It constructs and analyzes a large class of new normal forms, highlighting their naturalness and differences from the Chern-Moser form, including simpler variants without the trace operator.

Findings

New normal forms with various normalization conditions

Some forms are simpler than Chern-Moser, avoiding the trace operator

Provides a comprehensive, accessible exposition

Abstract

In this paper we construct a large class of new normal forms for Levi-nondegenerate real hypersurfaces in complex spaces. We adopt a general approach illustrating why these normal forms are natural and which role is played by the celebrated Chern-Moser normal form. The latter appears in our class as the one with the "maximum normalization" in the lowest degree. However, there are other natural normal forms, even with normalization conditions for the terms of the same degree. Some of these forms do not involve the cube of the trace operator and, in that sense, are simplier than the one by Chern-Moser. We have attempted to give a complete and self-contained exposition (including proofs of well-known results about trace decompositions) that should be accessible to graduate students.