Geometric flows and general relativity

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This project aims to unite the mathematics of geometric flows with the physics of gravitation. Geometric flows describe ways to deform manifolds. Manifolds are curved spaces, such as the familiar round sphere, but whereas the curvature of a round sphere is very regular and uniform, an arbitrary manifold can have bumps and valleys with quite general curvature, and can have any number of dimensions. In physics, curvature in the universe is the cause of gravitation, and so is central to general relativity as well as to string theory. An important mathematical development in recent years is the study of geometric flow equations. These are partial differential equations that specify ways to deform the curvature of a manifold, often making the manifold more uniform, though sometimes making it more curved. A singularlity forms if the curvature becomes infinitely large. The study of these equations has led to exciting developments in mathematics, such as the proof of the famous Poincaré conjecture and, more recently, the diffeomorphic ¼-pinched sphere theorem. But these flows also have a role to play in physics, and that aspect has been less well developed. One aim of this project is to develop the theory of these flows with a view to physics applications. One such application is the notion of mass in general relativity. There is a good definition of total mass of an isolated gravitating system, but there are several competing notions of the quasilocal mass of parts of a system, and each has drawbacks. One of the most useful notions of quasilocal mass was presented by Robert Bartnik, but it is one of the most difficult to calculate. The effort to overcome this difficulty led to certain conjectures which are important in their own right. This in turn leads to the consideration of geometric flows with boundaries. Very little is known about these flows, and this proposal is intended to change that. The methods I propose to develop will also have applications to numerical geometric flows and to the study of static and, perhaps, stationary systems in relativity, such as a star that may rotate but does not emit gravitational radiation, or a black hole in equilibrium with a "brane".