Research

Geometric Analysis, the mathematics required to understand gravity as the curvature of spacetime, and whose further development is essential for a deeper understanding of physics

Professor Bray's research in geometric analysis involves minimal surfaces, scalar curvature, conformal geometry, null geometry, geometric flows, and harmonic functions, among other topics. In his thesis, he proved a volume comparison theorem for scalar curvature which generalizes Bishop's theorem (for Ricci curvature) in three dimensions. Following an idea proposed by Richard Schoen, Andre Neves and he were the first to compute the Yamabe invariant of RP^3, enabling them to completely classify 3-manifolds with greater Yamabe invariant. With Simon Brendle, Michael Eichmair, and Andre Neves, Professor Bray proved a variety of rigidity theorems for compact manifolds, relating scalar curvature to the area of minimal surfaces.

In addition, Professor Bray's proof of the Riemannian Penrose Conjecture in dimension 3, and in dimensions less than 8 with Dan Lee, is a very interesting result about the relationship between scalar curvature and minimal surfaces on asymptotically flat manifolds. This result may be viewed entirely in geometric terms, with the happy coincidence that it also solved a major open problem about black holes, conjectured by Sir Roger Penrose in 1973.

More generally, most of Professor Bray's work, including his research in General Relativity and ideas relating to Dark Matter, may be thought of as topics in Geometric Analysis. If you are a student at Duke, consider taking Math 421, undergraduate differential geometry, with Professor Bray, followed by Math 621, graduate differential geometry. These courses will give you the background you need to use differential geometry as a tool for understanding general relativity and the large-scale structure of the universe.

This article describes Professor Bray's work in 2014 on the Hawking mass and time flat surfaces in spacetimes, joint with Jauregui and Mars.