Time Flat Curves and Surfaces, Geometric Flows, and the Penrose Conjecture

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AbstractAward: DMS 1406396, Principal Investigator: Hubert L. BrayThis project aims to continue to advance our understanding of the implications of Einstein's theory of general relativity. This theory is at the foundation of our understanding of gravity and has applications in everyday use, such as global positioning system (GPS) technology, now present in most smart phones. General relativity also unifies the notions of space, time, mass, and energy and provides the framework for understanding supermassive black holes and the Big Bang, the most energetic event in the history of the universe. More specifically, this project focuses on finding a better understanding of the notion of mass in general relativity, whether it relates to a black hole or to defining the mass of an exploding supernova. Since geometric analysis is the field of mathematics required to precisely state general relativity, this project will use many techniques from geometric analysis to study these questions.Even more specifically, this project has two main research directions. The first is motivated by the desire to understand the geometry and mass of surfaces in spacetimes. Jeff Jauregui, Marc Mars, and the PI have found that the Hawking mass of a surface is nondecreasing under a new geometric flow called "uniformly area expanding time flat flow." The key idea is that the Hawking mass can overestimate the mass of a region inside a surface if the surface is not "time flat," which we define precisely. A time flat surface is a generalization of the concept of a surface contained in a constant time slice of a static spacetime. Uniformly area expanding time flat flow, which is interesting by itself, grows the area form of the surface at a constant rate while keeping the surface, as a whole, time flat. Existence, uniqueness, and asymptotics of this flow are very important questions to study in a variety of contexts. The second research direction, which is related to the first, is to understand the mass of black holes as they relate to the Penrose conjecture, open since 1973. While the PI proved the Riemannian Penrose conjecture for any number of black holes in 1999, which is the case when the hypersurface of a spacetime has zero second fundamental form, the general case pertaining to any slice of a spacetime is still open. While these questions can be posed in the language of physics, they are also important statements relating to scalar and Ricci curvature, minimal surfaces and mean curvature, isoperimetric regions, connections on bundles, submanifold geometry, geometric flows, and geometric partial differential equations.