On long-time existence for the flow of static metrics with rotational symmetry

TL;DR

This paper investigates the long-time behavior of a geometric flow related to static Einstein metrics with rotational symmetry, showing conditions for eternal existence and analyzing singularity formation.

Contribution

It provides the first analysis of List's flow under SO(n) symmetry with asymptotic flatness, establishing criteria for long-time existence and singularity development.

Findings

Flow is immortal under certain regularity conditions.

Rescaling near singularities yields ancient Ricci flows.

Flow analysis connects to black hole thermodynamics and quasi-local mass.

Abstract

B List has proposed a geometric flow whose fixed points correspond to solutions of the static Einstein equations of general relativity. This flow is now known to be a certain Hamilton-DeTurck flow (the pullback of a Ricci flow by an evolving diffeomorphism) on RxM^n. We study the SO(n) rotationally symmetric case of List's flow under conditions of asymptotic flatness. We are led to this problem from considerations related to Bartnik's quasi-local mass definition and, as well, as a special case of the coupled Ricci-harmonic map flow. The problem also occurs as a Ricci flow with broken SO(n+1) symmetry, and has arisen in a numerical study of Ricci flow for black hole thermodynamics. When the initial data admits no minimal hypersphere, we find the flow is immortal when a single regularity condition holds for the scalar field of List's flow at the origin. This regularity condition can be shown to hold at least for n=2. Otherwise, near a singularity, the flow will admit rescalings which converge to an SO(n)-symmetric ancient Ricci flow on R^n.