Vol. 255, No. 1, 2012

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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
Abstract

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

-∂gij = − 2Rij, t ∈ [0,T ),
∂t

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
Authors
Nam Q. Le
Department of Mathematics
Columbia University
New York NY 10027
United States
Natasa Sesum
Department of Mathematics
Rutgers University
Piscataway NJ 10027
United States
http://www.math.rutgers.edu/~natasas