On the long-time behavior of Ricci flow and Ricci flow surgery

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

A Ricci flow is a geometric process that can be used to smooth out, and sometimes homogenize, a given space. Its mathematical significance has become apparent by the fact that it could be used to prove various conjectures, such as the Poincaré and Geometrization Conjectures in 3-dimensional spaces. A general expectation in the study of Ricci flows is that the flow produces a geometry in the limit that is somehow inherent to the topology, i.e. the loose makeup, of the underlying space. Most often, however, the flow develops certain singularities, which have to be removed by so-called "surgeries" before the flow can be continued. Despite their powerful topological implications, Ricci flows with surgery are still not well understood in dimensions 3 or higher. The goal of this project is to obtain a better understanding of the long-time behavior of 3 dimensional Ricci flows with surgery, and the dependence of the evolved geometries on initial conditions. Moreover, the study of Ricci flows in higher dimensions is suggested. The proposal is split into three projects. The first project concerns the analysis of the long-time behavior of 3 dimensional Ricci flows with surgery. This project builds on previous work of the principal investigator, in which the finiteness of the number of surgeries was established and in which an initial description of the flow's long-time asymptotics was derived. The objective of the second project is to construct continuous families of Ricci flows with surgery, starting from a given continuous family of Riemannian metrics. In such families, surgeries may move continuously in space and time depending on the parameter, and they may appear or disappear. A successful construction of such families can most likely be used to solve a conjecture that states that the space of positive scalar curvature metrics on the 3-sphere is contractible. Moreover, it may be used to solve the Generalized Smale Conjecture, which classifies the topology of diffeomorphism groups of spherical 3-manifolds. In the third project, the principal investigator proposes the work on several problems associated with the study of Ricci flows with bounded scalar curvature. This study is a continuation of previous work conducted in collaboration with Qi Zhang. The suggested problems include the analysis of singularities in 4-dimensional Ricci flows with bounded scalar curvature, and the study of non-collapsed, long-time existent Ricci flows, especially in dimension 4.