On three-dimensional Type I κ-solutions to the Ricci flow

Yongjia Zhang

Paper · Aug 8, 2017

On three-dimensional Type I $κ$ -solutions to the Ricci flow

Yongjia Zhang

math.DGarXiv:1708.02341v2

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Abstract

In this short note, we prove that the only simply connected noncompact three-dimensional Type I $κ$ -solution to the Ricci flow is the shrinking cylinder. This work can be regarded as a generalization of Cao and Chow, and a complement of Ding and Ni. Up to this point, three-dimensional $κ$ -solutions of Type I are completely classified, and it remains interesting to work further towards Perelman's assertion, that the only remaining possibility of three-dimensional noncompact $κ$ -solution is the Bryant soliton. Brendle is working to that end. The classification of three-dimensional $κ$ -solution is of importance to the study of four-dimensional Ricci flows, because of a possible dimension-reduction procedure.

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Submitted: Aug 8, 2017
Last revised: Feb 20, 2018 (v2)
Subjects: math.DG

Ricci flow Ancient solutions Singularities 3-manifolds 4-manifolds Boundaries & non-compact

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

12B.3
The Structure of κ-SolutionsBook 12B · κ-Solutions and Surgery
Necks and caps, compactness, and universal estimates.
11B.4
Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities
Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.
11B.6
Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities
The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.

[1] 12B.3
The Structure of κ-SolutionsBook 12B · κ-Solutions and Surgery
Necks and caps, compactness, and universal estimates.

[2] 11B.4
Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities
Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

[3] 11B.6
Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities
The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.

On three-dimensional Type I $κ$ -solutions to the Ricci flow

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