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

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.