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

B List has proposed a geometric flow whose fixed pointscorrespond to solutions of the static Einstein equations of generalrelativity. This flow is now known to be a certain Hamilton–DeTurckflow (the pullback of a Ricci flow by an evolving diffeomorphism) on${\mathbb R}\times M^n$. We study the ${\rm SO}(n)$ rotationallysymmetric case of List’s flow under conditions of asymptoticflatness. We are led to this problem from considerations related toBartnik’s quasi-local mass definition and, as well, as a specialcase of the coupled Ricci-harmonic map flow. The problem also occursas a Ricci flow with broken ${\rm SO}(n+1)$ symmetry, and has arisenin a numerical study of Ricci flow for black hole thermodynamics.When the initial data admits no minimal hypersphere, we find theflow is immortal when a single regularity condition holds for thescalar field of List’s flow at the origin. This regularity conditioncan be shown to hold at least for $n=2$. Otherwise, near asingularity, the flow will admit rescalings which converge to an${\rm SO}(n)$-symmetric ancient Ricci flow on ${\mathbb R}^n$.

Full Text (PDF format)

Published 1 January 2010