Geometry Forum
Friday, November 13, 2009, 1:15pm, 205 Physics
George Lam (Duke University)
The Integral Riemannian Penrose Inequality

Abstract:

Let (M,g) be a complete, asymptotically flat Riemannian 3-manifold with nonnegative scalar curvature. If (M,g) has an outer minimizing horizon $\Sigma$ with area A, then the Riemannian Penrose Inequality states that the ADM mass m of (M,g) satisifies \[m \geq \sqrt{A/16\pi},\] with equality if and only if (M,g) is isometric to the Schwarzschild metric. A careful analysis of the two known proofs of the inequality, due to Huisken-Illmanen (for one black hole) and Bray (multiple black holes), reveal that we actually have inequalities of the form \[m \geq \sqrt{A/16\pi} + \int_M RQ dV,\] where R is the scalar curvautre of (M,g) and Q is a nonnegative 'potential'. I will give a brief overview of how to obtain such inequalities using inverse mean curvautre flow of surfaces and conformal flow of metrics used in the two aforementioned proofs of the Riemannian Penrose Inequality. I will also discuss some ideas as to how one may attempt to find other such Q.

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