Paper · Sep 15, 2020

A note on convergence of noncompact nonsingular solutions of the Ricci flow

Qi S Zhang

math.DGarXiv:2009.06892
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Abstract

We extend some convergence results on nonsingular compact Ricci flows in the papers, and to certain infinite volume noncompact cases which are "partially" nonsingular. As an application, for a finite time singularity which is partially type I, it is shown that a blow up limit is a gradient shrinking soliton.

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Submitted
Sep 15, 2020
Subjects
math.DG
Ricci flowRicci solitonsSingularitiesBoundaries & non-compact

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper New Volume Comparison results and Applications to degeneration of Riemannian metrics (Qi S Zhang and Meng Zhu, 2016)2Zhang 2016 Minimizers of the sharp Log entropy on manifolds with non-negative Ricci curvature and flatness (Qi S Zhang, 2017)3Zhang 2017 On the κ-solutions of the Ricci flow on noncompact 3-manifolds (Liang Cheng and Anqiang Zhu, 2018)1Cheng 2018 Ancient solutions to the Ricci flow in dimension 3 (S. Brendle, 2018)4Brendle 2018 Ricci Solitons, Conical Singularities, and Nonuniqueness (Sigurd B. Angenent and Dan Knopf, 2019)6Angenent 2019 Closed Ricci Flows with Singularities Modeled on Asymptotically Conical Shrinkers (Maxwell Stolarski, 2022)5Stolarski 2022 A rigidity result for ancient Ricci flows (Qi S. Zhang, 2024)8Zhang 2024 A note on Huisken monotonicity-type formula for the mean curvature flow in a gradient shrinking extended Ricci soliton background (José N. V. Gomes, Matheus Hudson and Hikaru Yamamoto, 2024)7Gomes 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the solutions of the Ricci flow on noncompact 3-manifolds

    Liang Cheng and Anqiang Zhu

    Topics Boundaries & non-compact, Singularities · Shared terms “partially”, “finite time”

  2. New Volume Comparison results and Applications to degeneration of Riemannian metrics

    Qi S Zhang and Meng Zhu

    Topics Singularities, Ricci solitons · Same author Qi S Zhang

  3. Minimizers of the sharp Log entropy on manifolds with non-negative Ricci curvature and flatness

    Qi S Zhang

    Topics Boundaries & non-compact · Same author Qi S Zhang

  4. Ancient solutions to the Ricci flow in dimension 3

    S. Brendle

    Topics Boundaries & non-compact, Singularities · Shared terms “time singularity”, “finite time”

  5. Closed Ricci Flows with Singularities Modeled on Asymptotically Conical Shrinkers

    Maxwell Stolarski

    Topics Boundaries & non-compact, Singularities, Ricci solitons · Shared terms “time singularity”, “finite time”

  6. Ricci Solitons, Conical Singularities, and Nonuniqueness

    Sigurd B. Angenent and Dan Knopf

    Topics Boundaries & non-compact, Singularities, Ricci solitons · Shared terms “shrinking soliton”, “gradient shrinking”

  7. A note on Huisken monotonicity-type formula for the mean curvature flow in a gradient shrinking extended Ricci soliton background

    José N. V. Gomes, Matheus Hudson and Hikaru Yamamoto

    Topics Boundaries & non-compact, Ricci solitons · Shared terms “gradient shrinking”

  8. A rigidity result for ancient Ricci flows

    Qi S. Zhang

    Topics Singularities · Same author Qi S Zhang

Read around it

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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

In the guidebook

  1. 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.

  2. 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.

  3. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.