Special Feature: The Takagi Lectures

Japanese Journal of Mathematics

, Volume 6, Issue 1, pp 45-61

Evolution equations in Riemannian geometry

  • Simon BrendleAffiliated withDepartment of Mathematics, Stanford University Email author 

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Abstract

A fundamental question in Riemannian geometry is to find canonical metrics on a given smooth manifold. In the 1980s, R.S. 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.

Keywords and phrases

53C44 (primary) 35K55 53C21 58J35 (secondary)

Mathematics Subject Classification (2010)

scalar curvature sectional curvature Yamabe problem Ricci flow Sphere Theorem