Communications in Analysis and Geometry

Volume 18 (2010)

Number 4

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

Pages: 705 – 741



Liljana Gulcev (Department of Mathematical and Statistical Sciences, University of Alberta, Canada)

Todd A. Oliynyk (School of Mathematical Sciences, Monash University, Clayton, Australia)

Eric Woolgar (Department of Mathematical and Statistical Sciences, University of Alberta, Canada)


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 ${\mathbb R}^n$.

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