# Rigid Levi degenerate hypersurfaces with vanishing CR-curvature

## TL;DR

This paper classifies a specific class of rigid hypersurfaces in complex three-dimensional space with certain degeneracy and curvature properties, revealing their structure through solutions to classical differential equations.

## Contribution

It provides a complete description of rigid Levi degenerate hypersurfaces with zero CR-curvature, removing previous restrictive assumptions and linking their structure to well-known differential equations.

## Key findings

- Classification of hypersurfaces via differential equations
- Connection to conformal metrics with negative curvature
- Explicit description of the hypersurfaces' structure

## Abstract

We continue our study, initiated in an earlier article, of a class of rigid hypersurfaces in ${\mathbb C}^3$ that are 2-nondegenerate and uniformly Levi degenerate of rank 1, having zero CR-curvature. We drop the restrictive assumptions of the earlier paper and give a complete description of the class. Surprisingly, the answer is expressed in terms of solutions of several well-known differential equations, in particular, the equation characterizing conformal metrics with constant negative curvature and a nonlinear $\bar\partial$-equation.