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

Research output: Contribution to journalArticle

17 Citations (Scopus)

Abstract

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.

Original languageEnglish (US)
Pages (from-to)487-510
Number of pages24
JournalGeometry and Topology
Volume7
StatePublished - 2003
Externally publishedYes

Fingerprint

Ricci Flow
Injectivity
Radius
n-dimensional
Curvature
Estimate
Subsequence
Metric space
Riemannian Manifold
Converge
Derivative
Metric
Theorem

Keywords

  • Gromov-hausdor convergence
  • Ricci flow

ASJC Scopus subject areas

  • Geometry and Topology

Cite this

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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.

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