Scalar Curvature, Geometric Flows, and the General Penrose Conjecture
DMS - 0206483 Hubert Bray, MIT
The primary goal of this research is to prove the full Penrose conjecture in general relativity about the mass of black holes in a spacetime. In 3+1 dimensions, this conjecture states that the total mass of a spacetime with nonnegative energy density everywhere is greater than or equal to the square root of the total area of the event horizons of all of the black holes in the spacetime divided by 16 pi. This conjecture can be thought of as stating that the mass contributed by a collection of black holes is at least the square root of their surface areas divided 16 pi, so that nonnegative energy density everywhere else in the universe forces the total mass of the spacetime to be at least this amount. The Penrose conjecture is best thought of as a conjecture on arbitrary three dimensional space-like slices of the spacetime. In the special case that the space-like slice is assumed to have zero second fundamental form, the conjecture is known as the Riemannian Penrose conjecture. This conjecture was first proved for a single black hole by Huisken and Ilmanen in 1997 and then for any number of black holes by the author in 1999. The author would also like to prove the Riemannian Penrose Conjecture in dimensions higher than three and is close to announcing this result for dimensions less than eight. Dimensions eight and higher present additional geometric and analytical challenges arising from the fact that the apparent horizons of black holes manifest themselves as minimal hypersurfaces which can have co-dimension seven singularities. The positive mass theorem is also still open in these dimensions for similar reasons, and is another very interesting related problem.
The motivation for the above problems is to gain a better understanding of General Relativity. While Einstein's theory of General Relativity is experimentally the best known theory of gravity, their are many theoretical questions about the theory which are not well understood at all. For example, given the universe at some initial time, do the Einstein equations have unique well-behaved solutions in the future as one would hope, or do singularities typically occur which might radiate or consume energy for example? The theory also predicts the existence of black holes, and astronomers believe that they have been able to detect the location of many black holes including one which is 3 million times the mass of the sun at the center of our galaxy. Since doing experiments with black holes is currently not feasible, it makes since to understand this potentially very important phenomenon on a theoretical level for now. This research project hopes to lead to a better understanding of black holes as well as energy and mass in General Relativity.
