Analysis of Singularities of the Ricci Flow

University Of Connecticut, Storrs CT

Investigators

Abstract

Differential geometry is the key mathematics in Einstein's theory of relativity. Indeed, Einstein wrote his famous field equations in the language of tensors on manifolds, and these equations have since been well studied by mathematicians and physicists alike. Certainly, besides being fundamental in general relativity, differential geometry is also very useful in other fields of science, such as in control theory, in computer vision, data analysis, and many others. For this reason, understanding the structure of manifolds is a fundamental problem in science. This project will focus on the behavior of geometric flows on manifolds. A typical example of a geometric flow is the heat equation, which describes the distribution of heat in a region over time. This proposal studies a more advanced form of the heat equation, called the Ricci flow. A Riemannian metric on a manifold tells us about the shape of that object, how to measure angles and distances. The Ricci flow is a heat-type equation for Riemannian metrics. It is hoped, and confirmed in some cases, that the Ricci flow will evolve a given metric on a manifold to an improved one, such as an Einstein metric. However, a major difference between this theory and that of the standard heat equation is that the Ricci flow is a non-linear equation, and as such it usually develops singularities after some time. When such singularities are understood, the process may be continued. This has played a central role in the proof of the long-standing Poincare conjecture about the topology of three dimensional manifolds. The main goal of this project is to understand such singularities in dimension four, and to investigate the implications of our findings to the structure of four dimensional manifolds. Because Ricci flow can be seen as the renormalization group flow in string theory, there are other possible applications of this study to theoretical physics. Other related flows, like the mean curvature flow, have further remarkable applications to other fields, such as in computer visualization, for eliminating noise, or in metallurgy, for heat treatment of metals. The outreach components of this project disseminate the results to general public and contribute to the development of young talent. Ricci flow was introduced by Richard Hamilton in the early eighties, in a fundamental work devoted to understanding positively curved three dimensional manifolds. It became clear later that if one flows an arbitrary metric on a given manifold, the flow will generally develop singularities. One needs to understand these singularities in order to continue the flow, and to not lose any significant topological information about the space. The singularities of Ricci flow are modeled by Ricci solitons, which are fixed points of the flow, modulo diffeomorphisms and scaling. Three-dimensional shrinking Ricci solitons have been classified through the work of Hamilton, Ivey and Perelman. This has important consequences to understanding the behavior of Ricci flow with surgeries on three-dimensional manifolds, and indeed, for the resolution of the Poincare conjecture. The main goal of this project is to classify four-dimensional complete noncompact Ricci solitons. This will be achieved through a complete understanding of the asymptotic geometry of these spaces and through studying corresponding rigidity questions. It is expected that this project will advance our insight on the behavior of Ricci flow in dimension four, which will enable a Ricci flow approach to some important questions about the topology of four dimensional manifolds. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.