Ngô Quốc Anh

September 5, 2011

The Riemannian Penrose inequality

Filed under: Riemannian geometry — Ngô Quốc Anh @ 1:43

In mathematical general relativity, the Penrose inequality, first conjectured by Sir Roger Penrose, estimates the mass of a spacetime in terms of the total area of its black holes and is a generalization of the positive mass theorem. The Riemannian Penrose inequality is the most important special case. Specifically, if (M, g) is an asymptotically flat Riemannian 3-manifold with nonnegative scalar curvature and ADM mass m, and A is the area of the outermost minimal surface (possibly with multiple connected components), then the Riemannian Penrose inequality asserts

\displaystyle m \geq \sqrt{\frac{A}{16\pi}}.

This is purely a geometrical fact, and it corresponds to the case of a complete three-dimensional, space-like, totally geodesic submanifold of a (3 + 1)-dimensional spacetime. Such a submanifold is often called a time-symmetric initial data set for a spacetime. The condition of (M, g) having nonnegative scalar curvature is equivalent to the spacetime obeying the dominant energy condition.

This inequality was first proved by Gerhard Huisken and Tom Ilmanen in 1997 [here and here] in the case where A is the area of the largest component of the outermost minimal surface. Their proof relied on the machinery of weakly defined inverse mean curvature flow, which they developed. In 1999, Hubert Bray [here] gave the first complete proof of the above inequality using a conformal flow of metrics. Both of the papers were published in 2001 in the Journal of Differential Geometry.

Source: Wiki

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