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On Hamilton’s Ricci flow and Bartnik’s construction of metrics of prescribed scalar curvature

Connections for Women: Mathematical General Relativity September 03, 2013 - September 04, 2013

September 03, 2013 (04:00 PM PDT - 04:30 PM PDT)
Speaker(s): Chen-Yun Lin (University of Connecticut)
Location: SLMath: Eisenbud Auditorium
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video

v1154

Abstract Riemannian 3-manifolds with prescribed scalar curvature arise naturally in general relativity as spacelike hypersurfaces in the underlying spacetime. In 1993, Bartnik introduced a quasi-spherical construction of metrics of prescribed scalar curvature on 3-manifolds. This quasi-spherical ansatz has a background foliation with round metrics and converts the problem into a semi-linear parabolic equation. It is also known by work of R. Hamilton and B. Chow that the evolution under the Ricci flow of an arbitrary initial metric $g_0$ on $S^2$, suitably normalized, exists for all time and converges to the round metric. In this talk, we describe a construction of metrics of prescribed scalar curvature using solutions to the Ricci flow. Considering background foliations given by Ricci flow solutions, we obtain a parabolic equation similar to Bartnik’s. We discuss conditions on the scalar curvature that guarantees the solvability of the parabolic equation, and thus the existence of asymptotically flat 3-metrics with a prescribed inner boundary. In particular, many examples of asymptotically flat 3-metrics with outermost minimal surfaces are obtained
Supplements
18949?type=thumb Lin 176 KB application/pdf Download
Video/Audio Files

v1154

H.264 Video v1154.m4v 166 MB video/mp4 rtsp://videos.msri.org/data/000/017/699/original/v1154.m4v Download
Quicktime v1154.mov 234 MB video/quicktime rtsp://videos.msri.org/data/000/017/700/original/v1154.mov Download
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