Hyperbolic mass via horospheres

TL;DR

This paper develops geometric formulas for the mass of asymptotically hyperbolic manifolds using horospheres, leading to new rigidity results that characterize hyperbolic space under certain curvature and boundary conditions.

Contribution

It introduces a novel approach to compute mass via coordinate horospheres and establishes new rigidity theorems for hyperbolic manifolds based on these formulas.

Findings

Derived explicit geometric formulas for mass using horospheres.

Proved a new rigidity theorem for hyperbolic space with scalar curvature bounds.

Identified regions near infinity that do not affect the mass, improving rigidity results.

Abstract

We derive geometric formulas for the mass of asymptotically hyperbolic manifolds using coordinate horospheres. As an application, we obtain a new rigidity result of hyperbolic space: if a complete asymptotically hyperbolic manifold has scalar curvature lower bound -n(n-1) and is isometric to hyperbolic space outside a coordinate horosphere, then the manifold is isometric to hyperbolic space. In addition, we apply our formula to investigate regions near infinity that do not contribute to the mass quantity, which leads to improved rigidity results of hyperbolic space.