Geometry & Topology 7 (2003) 487–510

DOI: 10.2140/gt.2003.7.487

arXiv: math.DG/0211191

Abstract

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

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