A d'Alembert Formula for Hopf Hypersurfaces

TL;DR

This paper derives an explicit formula for Hopf hypersurfaces in complex hyperbolic space using a generalized Gauss map, enabling their reconstruction from the map's image, especially when the principal curvature is small.

Contribution

It introduces a generalized Gauss map approach to characterize and reconstruct Hopf hypersurfaces with small principal curvature in complex hyperbolic space.

Findings

Explicit formula for Hopf hypersurfaces derived

Hypersurfaces can be reconstructed from the Gauss map image

Applicable to hypersurfaces with small principal curvature

Abstract

A Hopf hypersurface in complex hyperbolic space CH^n is one in which the complex structure applied to the normal vector is a principal direction at each point. In this paper, Hopf hypersurfaces for which the corresponding principal curvature is small (relative to the ambient sectional curvature) are studied by means of a generalized Gauss map into a product of spheres, and it is shown that the hypersurface may be recovered from the image of this map, via an explicit formula.