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

TL;DR

This paper investigates the stability of asymptotically hyperbolic static solutions to Einstein's equations with negative cosmological constant, using Ricci-harmonic flow and a new entropy variant.

Contribution

It establishes stability criteria for static solutions under Ricci-harmonic flow and introduces a novel expander entropy for this flow.

Findings

Static metrics are stable if and only if a positive mass theorem holds.

A new variant of expander entropy for Ricci-harmonic flow is developed.

The paper links stability to geometric and energetic properties of the solutions.

Abstract

We prove dynamical stability and instability theorems for asymptotically hyperbolic static solutions of Einstein's equation with $\Lambda<0$, 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.