Geometric Analysis and Topology Seminar

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Isoperimetric Structure of Initial Data Sets

Speaker: Michael Eichmair, MIT

Location: Warren Weaver Hall 201

Date: Friday, November 4, 2011, 11 a.m.

Synopsis:

I will present joint work with Jan Metzger. A basic question in mathematical relativity is how geometric properties of an asymptotically flat manifold (or initial data set) encode information about the physical properties of the space time that it is embedded in. For example, the square root of the area of the outermost minimal surface of an initial data with non-negative scalar curvature provides a lower bound for the "mass" of its associated space time, as was conjectured by Penrose and proven by Bray and Huisken-Ilmanen. Other special surfaces that have been studied in this context include stable constant mean curvature surfaces and isoperimetric surfaces. I will explain why positive mass works to the effect that large stable constant mean curvature surfaces are always isoperimetric. This answers an old conjecture of Bray's and complements the results by Huisken-Yau and Qing-Tian on the "global uniqueness problem for stable CMC surfaces" in initial data sets with positive scalar curvature. Time permitting, I will sketch applications related to G. Huisken's isoperimetric mass and very recent related results with S. Brendle on further isoperimetric boundaries in the exact spatial Schwarzschild metric.