TY - JOUR

T1 - On long-time existence for the flow of static metrics with rotational symmetry

AU - Gulcev, Liljana

AU - Oliynyk, Todd

AU - Woolgar, Eric

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

UR - http://www.intlpress.com/CAG/2010/18-4/CAG-18-4-A3-gulcev.pdf

M3 - Article

VL - 18

SP - 705

EP - 741

JO - Communications in Analysis and Geometry

JF - Communications in Analysis and Geometry

SN - 1019-8385

IS - 4

ER -