The GBC mass for asymptotically hyperbolic manifolds

TL;DR

This paper introduces a new higher order mass for asymptotically hyperbolic manifolds using Gauss-Bonnet curvature, proves its positivity, and establishes related geometric inequalities, advancing the understanding of mass and inequalities in hyperbolic geometry.

Contribution

It defines the Gauss-Bonnet-Chern mass for asymptotically hyperbolic manifolds, proves its positivity, and links it to Penrose and Alexandrov-Fenchel inequalities in hyperbolic space.

Findings

Positive mass theorem for the new mass in hyperbolic graphs

Establishment of weighted Alexandrov-Fenchel inequalities in hyperbolic space

Derivation of an optimal Penrose type inequality under energy conditions

Abstract

The paper consists of two parts. In the first part, by using the Gauss-Bonnet curvature, which is a natural generalization of the scalar curvature, we introduce a higher order mass, the Gauss-Bonnet-Chern mass $m^{\H}_k$, for asymptotically hyperbolic manifolds and show that it is a geometric invariant. Moreover, we prove a positive mass theorem for this new mass for asymptotically hyperbolic graphs and establish a relationship between the corresponding Penrose type inequality for this mass and weighted Alexandrov-Fenchel inequalities in the hyperbolic space $\H^n$. In the second part, we establish these weighted Alexandrov-Fenchel inequalities in $\H^n$ for any horospherical convex hypersurface $\Sigma$. As an application, we obtain an optimal Penrose type inequality for the new mass defined in the first part for asymptotically hyperbolic graphs with a horizon type boundary $\Sigma$, provided that a dominant energy condition $\tilde L_k\ge0$ holds. Both inequalities are optimal.