# Nonlinear CR automorphisms of Levi degenerate hypersurfaces and a new gap phenomenon

## TL;DR

This paper classifies polynomial models of smooth real hypersurfaces in complex three-space that admit nonlinear CR automorphisms, establishing a new gap phenomenon and confirming a longstanding conjecture.

## Contribution

It provides a complete classification of models with nonlinear automorphisms and describes the possible dimensions of their automorphism Lie algebras, revealing a new gap phenomenon.

## Key findings

- Complete classification of polynomial models with nonlinear CR automorphisms
- Sharp 1-jet determination result for such hypersurfaces
- Identification of a new secondary gap in automorphism algebra dimensions

## Abstract

We give a complete classification of polynomial models for smooth real hypersurfaces of finite Catlin multitype in $\mathbb C^3$, which admit nonlinear infinitesimal CR automorphisms. As a consequence, we obtain a sharp 1-jet determination result for any smooth hypersurface with such model. The results also prove a conjecture of the first author about the origin of such nonlinear automorphisms (AIM list of problems, 2010). As another consequence, we describe all possible dimensions of the Lie algebra of infinitesimal CR automorphisms, which leads to a new "secondary" gap phenomenon.