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# 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

DOI: http://dx.doi.org/10.4310/CAG.2010.v18.n4.a3

#### Authors

#### Abstract

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 ${\mathbb R}^n$.