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
Email: glicken@math.ucsd.edu
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