Remarks on the curvature behavior at the first singular time of the Ricci flow
type: Item
instance of: scholarly article
title: Remarks on the curvature behavior at the first singular time of the Ricci flow
zbMATH Open document ID: 1241.53056
author: Nam Q. Le
author: Natasa Sesum
published in: Pacific Journal of Mathematics
publication date: 2012-05-02T00:00:00Z
full work available at URL: https://arxiv.org/abs/1005.1220
full work available at URL: p08.xhtml
Mathematics Subject Classification ID: 53C44
Mathematics Subject Classification ID: 35K10
zbMATH DE Number: 6029231
zbMATH Keywords: Ricci flow
zbMATH Keywords: scalar curvature
zbMATH Keywords: evolution equation
zbMATH Keywords: Gaussian shrinker
MaRDI profile type: Publication
OpenAlex ID: W2097209500
arXiv ID: 1005.1220
DOI: 10.2140/PJM.2012.255.155
review text: This paper discusses the behavior of the Ricci flow at the first singular time of one solution for the Ricci equation NEWLINE\[NEWLINE\frac{\partial}{\partial t} g_{ij}=-2R_{ij}NEWLINE\]NEWLINE on a smooth, compact and \(n\)-dimensional Riemannian manifold \(M\). If the flow has uniformly bounded scalar curvature and develops type I singularities at \(T\), it is shown that suitable blow-ups of the evolving metrics converge in the pointed Cheeger-Gromov sense to a Gaussian shrinker by using Perelman's \(\mathcal W\)-functional. If the flow has uniformly bounded scalar curvature and develops type II singularities at \(T\), it is shown that suitable scalings of the potential functions in Perelman's entropy functional converge to a positive constant on a complete, Ricci flat manifold. As a consequence, 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 \(T\).
description: scientific article; zbMATH DE number 6029231