Paper · Sep 17, 2019

Ricci Solitons, Conical Singularities, and Nonuniqueness

Sigurd B. Angenent, Dan Knopf

math.DGarXiv:1909.08087v2
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As on arXiv

Abstract

In dimension , there is a complete theory of weak solutions of Ricci flow - the singular Ricci flows introduced by Kleiner and Lott - which are unique across singularities, as was proved by Bamler and Kleiner. We show that uniqueness should not be expected to hold for Ricci flow weak solutions in dimensions . Specifically, we exhibit a discrete family of asymptotically conical gradient shrinking solitons, each of which admits non-unique forward continuations by gradient expanding solitons. (v2) We recast the Main Theorem in the language of Kleiner and Lott's Ricci Flow spacetimes and in addition show that topological nonuniqueness is possible for the solutions we construct.

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Submitted
Sep 17, 2019
Last revised
Mar 17, 2021 (v2)
Subjects
math.DG
Journal
Geom. Funct. Anal. (GAFA) 32 (2022), no.3, 411--489
DOI
10.1007/s00039-022-00601-y
Ricci flowRicci solitonsSingularitiesWeak & singular flowsStability & uniquenessBoundaries & non-compact

Its neighbourhood

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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 Perelman's Entropies for Manifolds with conical Singularities (Klaus Kroencke and Boris Vertman, 2019)4Kroencke 2019 Singularity formation of complete Ricci flow solutions (Timothy Carson, James Isenberg, Dan Knopf et al., 2020)2Carson 2020 Uniqueness of Asymptotically Conical Gradient Shrinking Solitons in G_2-Laplacian Flow (Mark Haskins, Ilyas Khan and Alec Payne, 2022)7Haskins 2022 Uniqueness of conical singularities for mean curvature flows (Tang-Kai Lee and Xinrui Zhao, 2023)6Lee 2023 Ricci flow from spaces with edge type conical singularities (Lucas Lavoyer, 2023)5Lavoyer 2023 Mean Curvature Flow from Conical Singularities (Otis Chodosh, J. M. Daniels-Holgate and Felix Schulze, 2023)8Chodosh 2023 Closed mean curvature flows with asymptotically conical singularities (Tang-Kai Lee and Xinrui Zhao, 2024)3Lee 2024 Infinite-dimensional dynamical instabilities of noncompact stationary Ricci flow solutions (Sigurd B. Angenent and Dan Knopf, 2025)1Angenent 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Infinite-dimensional dynamical instabilities of noncompact stationary Ricci flow solutions

    Sigurd B. Angenent and Dan Knopf

    Topics Boundaries & non-compact, Stability & uniqueness · Same authors Sigurd B. Angenent, Dan Knopf

  2. Singularity formation of complete Ricci flow solutions

    Timothy Carson, James Isenberg, Dan Knopf et al.

    Topics Boundaries & non-compact, Singularities, Stability & uniqueness · Same author Dan Knopf

  3. Closed mean curvature flows with asymptotically conical singularities

    Tang-Kai Lee and Xinrui Zhao

    Topics Weak & singular flows, Boundaries & non-compact, Singularities · Shared terms “conical singularities”, “asymptotically conical”

  4. Perelman's Entropies for Manifolds with conical Singularities

    Klaus Kroencke and Boris Vertman

    Topics Weak & singular flows, Singularities, Ricci solitons · Shared terms “conical singularities”

  5. Ricci flow from spaces with edge type conical singularities

    Lucas Lavoyer

    Topics Weak & singular flows, Singularities, Ricci solitons · Shared terms “conical singularities”

  6. Uniqueness of conical singularities for mean curvature flows

    Tang-Kai Lee and Xinrui ZhaoWider flows

    Topics Boundaries & non-compact, Singularities, Stability & uniqueness · Shared terms “conical singularities”, “asymptotically conical”

  7. Uniqueness of Asymptotically Conical Gradient Shrinking Solitons in G_2-Laplacian Flow

    Mark Haskins, Ilyas Khan and Alec Payne

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

  8. Mean Curvature Flow from Conical Singularities

    Otis Chodosh, J. M. Daniels-Holgate and Felix SchulzeWider flows

    Topics Weak & singular flows, Boundaries & non-compact, Singularities · Shared terms “conical singularities”, “asymptotically conical”

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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

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. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.