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Pseudolocality for the Ricci Flow and Applications

Open Access article
 Printed: Feb 2011
  • Albert Chau,
    Department of Mathematics, University of British Columbia, Vancouver, BC
  • Luen-Fai Tam,
    Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China
  • Chengjie Yu,
    Department of Mathematics, Shantou University, Shantou Guangdong, China
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Perelman established a differential Li--Yau--Hamilton (LYH) type inequality for fundamental solutions of the conjugate heat equation corresponding to the Ricci flow on compact manifolds. As an application of the LYH 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 flat manifolds. We also prove a long time existence result for the K\"ahler--Ricci flow on complete nonnegatively curved K\"ahler manifolds.
MSC Classifications: 53C44, 58J37, 35B35 show english descriptions Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
Perturbations; asymptotics
53C44 - Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
58J37 - Perturbations; asymptotics
35B35 - Stability

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