Explicit description of spherical rigid hypersurfaces in C^2

TL;DR

This paper explicitly characterizes all rigid hypersurfaces in C^2 equivalent to the Heisenberg sphere, using four real parameters, and connects their defining equations to solutions of a non-linear PDE related to Cartan curvature.

Contribution

It provides a complete explicit description of rigid hypersurfaces in C^2, linking geometric conditions to solutions of a specific non-linear PDE.

Findings

Rigid hypersurfaces are classified by four real parameters.

Explicit equations for these hypersurfaces are derived.

The equations solve a non-linear PDE related to Cartan curvature.

Abstract

We provide an explicit description of all rigid hypersurfaces that are equivalent to a Heisenberg sphere. These hypersurfaces are determined by 4 real parameters. The defining equations of the rigid spheres can also be viewed as the complete solution of a non-linear PDE that expresses the vanishing Cartan curvature condition for rigid hypersurfaces.