Precompactness of solutions to the Ricci flow in the absence of injectivity radius estimates

Consider a sequence of pointed $n$ –dimensional complete Riemannian manifolds ${(M_{i}, g_{i} (t), O_{i})}$ such that $t \in [0, T]$ are solutions to the Ricci flow and $g_{i} (t)$ 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

References

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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

Volume 7, issue 1 (2003)

David Glickenstein