#### Volume 7, issue 1 (2003)

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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
##### Abstract

Consider a sequence of pointed $n$–dimensional complete Riemannian manifolds $\left\{\left({M}_{i},{g}_{i}\left(t\right),{O}_{i}\right)\right\}$ such that $t\in \left[0,T\right]$ are solutions to the Ricci flow and ${g}_{i}\left(t\right)$ 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
Primary: 53C44
Secondary: 53C21