The geometry of Ricci solitons

University Of Connecticut, Storrs CT

Investigators

Abstract

Differential geometry is a branch of mathematics that studies the shapes of geometric objects, called manifolds. Differential geometry is the key mathematics in Einstein's theory of relativity, being used to describe the curvature of space-time in the presence of a body of mass and energy. As such, understanding the geometry and topology of manifolds is a fundamental problem in science. This project will focus on studying the behavior of evolution equations on manifolds. Geometric flows have proved to be very important in mathematics, in particular, Ricci flow has been very popular for its use in the resolution of some central problems, such as the long standing Poincare conjecture. Geometric flows can be used to study a fundamental question of geometry, which is to find canonical metrics on a given manifold. The Ricci flow proposes to do this in an analytic way, by flowing a given metric in time towards an improved, canonical one. This proposal will investigate the formation of singularities along the flow and will attempt to understand them in dimension four. There are possible applications of this study to theoretical physics, because Ricci flow can be seen as the renormalization group flow in string theory. Other related flows, like 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 developing of young talent. Ricci flow was introduced by 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 loose 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 scalings. Perelman classified three dimensional shrinking Ricci solitons, and used this classification in the resolution of the Poincare conjecture. This has attracted much attention on the higher dimensional problem, and its applications. The goal of this project is to understand the structure and properties of Ricci solitons, in arbitrary dimension. This will lead to a better understanding of how large the space of Ricci solitons is. The study will focus in particular on complete four dimensional shrinking Ricci solitons. There are significant new challenges to this problem, in particular, there exist higher dimensional examples of Ricci solitons which are not positively curved. This principal investigator will attempt to understand the asymptotic geometry of complete four dimensional shrinking Ricci solitons, which is key information for their ultimate classification.