Hypersurfaces with nonnegative Ricci curvature in hyperbolic space

TL;DR

This paper proves that complete, noncompact hypersurfaces with nonnegative Ricci curvature in hyperbolic space have at most two points at infinity, and if there are two, the hypersurface is a specific equidistant surface, answering a longstanding question.

Contribution

It establishes a classification of such hypersurfaces based on their asymptotic boundary, providing a rigidity result that characterizes equidistant hypersurfaces.

Findings

Hypersurfaces with nonnegative Ricci curvature have at most two points at infinity.

Two points at infinity imply the hypersurface is an equidistant surface.

The result confirms a question posed by Alexander and Currier in 1990.

Abstract

Based on properties of n-subharmonic functions we show that a complete, noncompact, properly embedded hypersurface with nonnegative Ricci curvature in hyperbolic space has an asymptotic boundary at infinity of at most two points. Moreover, the presence of two points in the asymptotic boundary is a rigidity condition that forces the hypersurface to be an equidistant hypersurface about a geodesic line in hyperbolic space. This gives an affirmative answer to the question raised by Alexander and Currier in 1990.