### 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) |
---|---|

Pages (from-to) | 487-510 |

Number of pages | 24 |

Journal | Geometry and Topology |

Volume | 7 |

State | Published - 2003 |

Externally published | Yes |

### Fingerprint

### Keywords

- Gromov-hausdor convergence
- Ricci flow

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

**Precompactness of solutions to the Ricci flow in the absence of injectivity radius estimates.** / Glickenstein, David A.

Research output: Contribution to journal › Article

*Geometry and Topology*, vol. 7, pp. 487-510.

}

TY - JOUR

T1 - Precompactness of solutions to the Ricci flow in the absence of injectivity radius estimates

AU - Glickenstein, David A

PY - 2003

Y1 - 2003

N2 - 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-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.

AB - 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-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.

KW - Gromov-hausdor convergence

KW - Ricci flow

UR - http://www.scopus.com/inward/record.url?scp=4243148519&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4243148519&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:4243148519

VL - 7

SP - 487

EP - 510

JO - Geometry and Topology

JF - Geometry and Topology

SN - 1465-3060

ER -