### Abstract

Consider a sequence of pointed n-dimensional complete Riemannian manifolds {(M_{i}, g_{i}(t), O_{i})} such that t ∈ [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-Hausdor 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.

Original language | English (US) |
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Pages (from-to) | 487-510 |

Number of pages | 24 |

Journal | Geometry and Topology |

Volume | 7 |

DOIs | |

State | Published - 2003 |

### Keywords

- Gromov-hausdor convergence
- Ricci flow

### ASJC Scopus subject areas

- Geometry and Topology