Static Vacuum Spacetimes with $\mathrm{\Lambda }<0$ as Attractors of the Ricci-Harmonic Flow

We prove dynamical stability and instability theorems for asymptotically hyperbolic static solutions of Einstein’s equation with $\mathrm{\Lambda }<0$, viewed as self-similar solutions of the Ricci-harmonic flow. More precisely, we show that static metrics are dynamically stable if and only if a positive mass type theorem holds for nearby metrics. Our key tool is a new variant of the expander entropy for the Ricci-harmonic flow.