Stephen McCormick: Variational Properties of the Volume-Renormalised Mass

Recently in joint work with Mattias Dahl and Klaus Kröncke we introduced a new mass-like invariant for asymptotically hyperbolic manifolds, which we call the volume-renormalised mass. In this talk I will first review some variational properties of the ADM mass for asymptotically flat manifolds before discussing some ongoing work regarding an analogue of these properties for the volume-renormalised mass. We first provide an alternate proof that extremisers of the volume-renormalised mass are exactly Einstein metrics, before arguing that this is no longer the correct characterisation of extremisers on a manifold with boundary (assuming appropriate boundary conditions) and discuss a generalised notion of Einstein metrics that should be considered instead.