Rasmus Johansen Jouttijärvi: Stability and PMT for Static Manifolds

In this talk we consider pairs (g,φ), where g is an asymptotically hyperbolic (AH) n-dimensional Riemannian metric and φ is a function satisfying a certain differential equation asymptotically. A subset of these are known as static pairs, as they may be combined to produce an (n+1)-dimensional static solution to Einsteins equations.

We introduce a renormalised entropy for AH pairs (g,φ), incorporating the AH version of the ADM mass. Static pairs serve as the only true critical points of the entropy, while several other semi-critical classes of pairs are considered. The gradient flow of the entropy is a modified Ricci-harmonic flow, and we shall aim to explain how the dynamical stability of static metrics under this flow is equivalent to a local positive mass theorem(PMT) for the aforementioned mass.

Joint work with Kröncke & Yudowitz. Inspired by their work on Poincaré-Einstein manifolds, and Dahl, Kröncke & McCormick’s work on a renormalised expander entropy in the AH setting.