Evolution equations in Riemannian geometry

TL;DR

This survey reviews evolution equations like Ricci and Yamabe flows in Riemannian geometry, highlighting their role in deforming metrics towards canonical forms and their application in proving the Differentiable Sphere Theorem.

Contribution

It provides a comprehensive overview of the global behavior of solutions to key evolution equations and their use in solving fundamental geometric problems.

Findings

Analysis of Ricci and Yamabe flow behaviors

Application of flows to prove the Differentiable Sphere Theorem

Insights into the long-term evolution of Riemannian metrics

Abstract

A fundamental question in Riemannian geometry is to find canonical metrics on a given smooth manifold. In the 1980s, R. Hamilton proposed an approach to this question based on parabolic partial differential equations. The goal is to start from a given initial metric and deform it to a canonical metric by means of an evolution equation. There are various natural evolution equations for Riemannian metrics, including the Ricci flow and the conformal Yamabe flow. In this survey, we discuss the global behavior of the solutions to these equations. In particular, we describe how these techniques can be used to prove the Differentiable Sphere Theorem. This article is based on the Takagi Lectures delivered by the author at the Research Institute for Mathematical Sciences, Kyoto University, on June 4, 2011.