Volume 7, issue 1 (2003)

Download this article
For printing
Recent Issues

Volume 24, 1 issue

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Ethics Statement
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Author Index
To Appear
Other MSP Journals
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


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.

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