Hypersurfaces in hyperbolic space with support function

TL;DR

This paper establishes a correspondence between convex hypersurfaces in hyperbolic space and conformal metrics on spheres, providing new insights into their geometric properties and solutions to elliptic problems.

Contribution

It introduces a global correspondence linking hypersurfaces and conformal metrics, and derives new reflection principles and Bernstein-type theorems for these geometric objects.

Findings

Injectivity conditions for the hyperbolic Gauss map

Unfolding immersed hypersurfaces into embedded ones

A Bernstein theorem for constant mean curvature hypersurfaces

Abstract

In this paper we develop a global correspondence between immersed horospherically convex hypersurfaces in hyperbolic space and complete conformal metrics on domains in the sphere. We establish results on when the hyperbolic Gauss map is injective and when an immersed horospherically convex hypersurface can be unfolded along the normal flow into an embedded one. These results allow us to establish general Alexandrov reflection principles of elliptic problems of both Weingarten hypersurfaces and complete conformal metrics and relations between them. Consequently, we are able to obtain, for instance, a strong Bernstein theorem for a complete, immersed, horospherically convex hypersurface of constant mean curvature in hyperbolic space.