Louis Yudowitz: Dynamical Stability and Instability of Poincare-Einstein Manifolds

A lot of work over the past decade or so has been devoted to proving the stability of compact Ricci solitons (of which Einstein manifolds are a special case). These solitons evolve self-similarly under Ricci flow and arise as critical points of certain geometric functionals. On the other hand, the non-compact case still poses issues, in large part due to a lack of suitable functionals. In this talk, we will see how to overcome this for Poincaré-Einstein manifolds (i.e. negative Einstein manifolds which are asymptotically hyperbolic) and develop a complete picture of their dynamical stability. In particular, stability will depend on the metric being a local maximizer/minimizer for a suitable (relative) entropy functional for asymptotically hyperbolic manifolds, which was recently introduced by Dahl-McCormick-Kröncke. Along the way, we will also prove a Łojasiewicz-Simon inequality for this entropy, which will serve as our main technical tool. Local maximizers of the entropy will also be shown to be equivalent to a local positive mass theorem and a volume comparison result. This is all based on recent joint work with Klaus Kröncke.