Remarks on the curvature behavior at the first singular time of the Ricci flow

Nam Q. Le and Natasa Sesum

Vol. 255 (2012), No. 1, 155–175

DOI: 10.2140/pjm.2012.255.155

Abstract

We study the curvature behavior at the first singular time of a solution to the Ricci flow

on a smooth, compact n-dimensional Riemannian manifold M. If the flow has uniformly bounded scalar curvature and develops Type I singularities at T, we show that suitable blow-ups of the evolving metrics converge in the pointed Cheeger–Gromov sense to a Gaussian shrinker by using Perelman’s 𝒲-functional. If the flow has uniformly bounded scalar curvature and develops Type II singularities at  T, we show that suitable scalings of the potential functions in Perelman’s entropy functional converge to a positive constant on a complete, Ricci flat manifold. We also show that if the scalar curvature is uniformly bounded along the flow in certain integral sense then the flow either develops a Type II singularity at T or it can be smoothly extended past time T.

Keywords

Ricci flow, scalar curvature, evolution

Mathematical Subject Classification 2010

Primary: 53C44

Secondary: 35K10

Milestones

Received: 7 October 2010

Revised: 23 February 2011

Accepted: 11 April 2011

Published: 14 March 2012