A Volume-Renormalized Mass for Asymptotically Hyperbolic Manifolds

Mattias Dahl, Klaus Kroencke, Stephen McCormick

Published 2023-07-12Version 1

We define a geometric quantity for asymptotically hyperbolic manifolds, which we call the volume-renormalized mass. It is essentially a linear combination of a renormalization of the volume and the standard ADM mass integral. We show that the volume-renormalized mass is well-defined and diffeomorphism invariant under weaker fall-off conditions than required to ensure the renormalized volume and ADM mass integral are well-defined separately. We prove several positivity results for this mass, and we use it to define a renormalized Einstein--Hilbert action and an expander entropy in the context of Ricci flow on asymptotically hyperbolic manifolds. Furthermore, we show that the expander entropy is monotonically nondecreasing under the Ricci flow, critical points are Poincar\'e--Einstein metrics, and local maximizers of the entropy are local minimizers of the volume-renormalized mass.