On Type I Singularities in Ricci flow

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. We also define a notion of density for Type I Ricci flows and use it to prove a regularity theorem reminiscent of White's partial regularity result for mean curvature flow.

13 pages, references added, final version, to appear in CAG

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• Structure theory of non-collapsed limits of Ricci flows
• Positively curved shrinking Ricci solitons are compact
• Ricci flow from spaces with isolated conical singularities
• Remarks on the extension of the Ricci flow
• Ricci-mean curvature flows in gradient shrinking Ricci solitons
• Remarks on curvature behavior at the first singular time of the Ricci flow
• Structure at infinity for shrinking Ricci solitons
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• Rotationally symmetric Ricci flow on ${\mathbb{R}}^{n+1}$
• Linear stability of compact shrinking Ricci solitons
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• A uniqueness theorem for asymptotically cylindrical shrinking Ricci solitons
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• On curvature estimates for four-dimensional gradient Ricci solitons
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• Local curvature estimates for the Laplacian flow
• Mean Value Inequalities and Conditions to Extend Ricci Flow
• Curvature pinching estimate under the Laplacian G_{2} flow
• Some extensions of the mean curvature flow in Riemannian manifolds
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• Long-time behavior of awesome homogeneous Ricci flows
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• A local curvature estimate for the Ricci flow
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• Convergence of the Ricci flow to Ricci-flat ALE manifolds and positive scalar curvature rigidity
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• Ends of (singular) Ricci shrinkers
• The sharp Li-Yau equality on Shrinking Ricci Solitons with applications
• Type of finite time singularities of the Ricci flow with bounded scalar curvature
• New proofs of Perelman's theorem on shrinking Breathers in Ricci flow
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• On the weighted forward reduced Entropy of Ricci flow
• Geometric properties of self-shrinkers in cylinder shrinking Ricci solitons
• The Rigidity of Ricci shrinkers of dimension four
• Ricci flow on homogeneous spaces with two isotropy summands
• Cotton tensor and conformal deformations of three-dimensional Ricci flow
• Scalar curvature behavior of homogeneous Ricci flows
• On a conjecture of Candelas and de la Ossa