On Type-I singularities in Ricci flow

2010/05/31 by Joerg Enders, Reto Müller, Peter M. Topping · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Curvature #Flow (mathematics) #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Gravitational singularity #Mathematical analysis #Mathematics #Pure mathematics #Ricci curvature #Ricci flow #Type (biology) #math.DG

paper · pdf · doi:10.4310/cag.2011.v19.n5.a4

published as Commun. Anal. Geom., Vol. 19, No. 5 (2011), 905-922 · 13 pages, references added, final version, to appear in CAG

openalex publication_date 2011/01/01 · arxiv created 2011/09/27 · arxiv updated 2015/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We define several notions of singular set for Type-I Ricci flows and show that they all coincide. In order to do this, we prove that blow-ups around singular points converge to nontrivial gradient shrinking solitons, thus extending work of Naber As a by-product we conclude that the volume of a finite-volume singular set vanishes at the singular time.

Citations

• Three-manifolds with positive Ricci curvature
• Gaussian densities and stability for some Ricci solitons
• Deforming the metric on complete Riemannian manifolds
• A Compactness Property for Solutions of the Ricci Flow
• Noncompact Shrinking 4-Solitons with Nonnegative Curvature
• Remarks on non-compact gradient Ricci solitons
• Remarks on curvature behavior at the first singular time of the Ricci flow

Cited by

• Ricci flow [wikipedia]

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