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Pseudolocality for the Ricci Flow and Applications

Published online by Cambridge University Press:  20 November 2018

Albert Chau ,
Luen-Fai Tam  and
Chengjie Yu
Albert Chau
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC email: [email protected]
Luen-Fai Tam
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China email: [email protected]
Chengjie Yu
Affiliation:
Department of Mathematics, Shantou University, Shantou Guangdong, China email: [email protected]
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Abstract

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Perelman established a differential Li-Yau-Hamilton $\left( \text{LHY} \right)$ type inequality for fundamental solutions of the conjugate heat equation corresponding to the Ricci flow on compact manifolds. As an application of the $\text{LHY}$ inequality, Perelman proved a pseudolocality result for the Ricci flow on compact manifolds. In this article we provide the details for the proofs of these results in the case of a complete noncompact Riemannian manifold. Using these results we prove that under certain conditions, a finite time singularity of the Ricci flow must form within a compact set. The conditions are satisfied by asymptotically flatmanifolds. We also prove a long time existence result for the Kähler-Ricci flow on complete nonnegatively curved Kähler manifolds.

Keywords

53C4458J3735B35
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 63 , Issue 1 , 01 February 2011 , pp. 55 - 85
Copyright
Copyright © Canadian Mathematical Society 2011

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