Volume 7, issue 1 (2003)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Precompactness of solutions to the Ricci flow in the absence of injectivity radius estimates

David Glickenstein

Geometry & Topology 7 (2003) 487–510

arXiv: math.DG/0211191

Abstract

Consider a sequence of pointed n–dimensional complete Riemannian manifolds {(Mi,gi(t),Oi)} such that t [0,T] are solutions to the Ricci flow and gi(t) have uniformly bounded curvatures and derivatives of curvatures. Richard Hamilton showed that if the initial injectivity radii are uniformly bounded below then there is a subsequence which converges to an n–dimensional solution to the Ricci flow. We prove a generalization of this theorem where the initial metrics may collapse. Without injectivity radius bounds we must allow for convergence in the Gromov–Hausdorff sense to a space which is not a manifold but only a metric space. We then look at the local geometry of the limit to understand how it relates to the Ricci flow.

Keywords
Ricci flow, Gromov–Hausdorff convergence
Mathematical Subject Classification 2000
Primary: 53C44
Secondary: 53C21
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Publication
Received: 9 December 2002
Accepted: 10 July 2003
Published: 29 July 2003
Proposed: Gang Tian
Seconded: John Morgan, Leonid Polterovich
Authors
David Glickenstein
Department of Mathematics
University of California
San Diego
9500 Gilman Drive
La Jolla
California 92093-0112
USA