Geometry & Topology, Vol. 7 (2003) Paper no. 14, pages 487--510.

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

David Glickenstein

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

AMS subject classification. Primary: 53C44. Secondary: 53C21.

DOI: 10.2140/gt.2003.7.487

E-print: arXiv:math.DG/0211191

Submitted to GT on 9 December 2002. Paper accepted 10 July 2003. Paper published 29 July 2003.

Notes on file formats

David Glickenstein
Department of Mathematics, University of California, San Diego
9500 Gilman Drive, La Jolla, CA 92093-0112, USA

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