A volume-renormalized mass for asymptotically hyperbolic manifolds

TL;DR

This paper introduces a new geometric quantity called the volume-renormalized mass for asymptotically hyperbolic manifolds, proving its well-definedness, invariance, and positivity, and exploring its applications in geometric analysis.

Contribution

It defines the volume-renormalized mass, establishes its invariance under weaker conditions, and connects it to entropy and Einstein-Hilbert action in geometric flows.

Findings

Volume-renormalized mass is well-defined and invariant.

Positivity results for the volume-renormalized mass.

Connections to entropy and Einstein-Hilbert action.

Abstract

We define a geometric quantity for asymptotically hyperbolic manifolds, which we call the volume-renormalized mass. It is essentially a linear combination of the ADM mass surface integral and a renormalization of the volume. We show that the volume-renormalized mass is well-defined and diffeomorphism invariant under weaker fall-off conditions than required to ensure that the renormalized volume and the ADM mass surface integral are well-defined separately. We prove several positivity results for the volume-renormalized mass. We also use it to define a renormalized Einstein--Hilbert action and a renormalized expander entropy which is nondecreasing under the Ricci flow. Further, we show that local maximizers of the entropy are local minimizers of the volume-renormalized mass.