Stephen McCormick, Luleå University of Technology
We introduce a volume-renormalised mass for asymptotically hyperbolic manifolds, which is essentially a linear combination of a renormalization of the volume and the standard ADM mass integral. This quantity is well-defined and diffeomorphism invariant under weaker fall-off conditions than required to ensure each term is finite separately, and exhibits several interesting properties. In addition to establishing some positive mass theorems, we use the volume-renormalised mass to define a normalised Einstein-Hiblert action and an expander entropy in the context of Ricci flow.
We show that this entropy is monotonically nondecreasing under the flow, critical points are negative Einstein metrics, and local maximisers of the entropy are local minimisers of the volume-renormalised mass.
This is joint work with Mattias Dahl and Klaus Kröncke.