The Penrose inequality and related topics

Brief background

The Penrose inequality, which gives a lower bound on the mass of a black hole in terms of the area of its horizon, is a conjecture which plays a central role in general relativity. A proof has been found in the Riemannian case by Huisken and Ilmanen and independently by Bray. The proofs are based on the study of monotone quantities under geometric flows, and similar techniques can be used to prove a range of fundamental inequalities. The general case of the Penrose inequality is still an open problem. Following an introduction to the Penrose inequality, I will discuss the methods which were used to prove it in the Riemannian case, their application to related problems, and some of the approaches which have been proposed to prove the general Penrose inequality.

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