Paper · Feb 6, 2019

Perelman's Entropies for Manifolds with conical Singularities

Klaus Kroencke, Boris Vertman

math.DGarXiv:1902.02097
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Abstract

In this paper we discuss Perelman's Lambda-functional, Perelman's Ricci shrinker entropy as well as the Ricci expander entropy on a class of manifolds with isolated conical singularities. On such manifolds, a singular Ricci de Turck flow preserving the isolated conical singularities exists by our previous work. We prove that the entropies are monotone along the singular Ricci de Turck flow. We employ these entropies to show that in the singular setting, Ricci solitons are gradient and that steady or expanding Ricci solitons are Einstein.

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Submitted
Feb 6, 2019
Subjects
math.DG, math.AP

Comments: 41 pages

Ricci flowRicci solitonsSingularitiesWeak & singular flowsHarmonic maps & DeTurck trick

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 Ricci de Turck flow on singular manifolds (Boris Vertman, 2016)3Vertman 2016 Ricci Solitons, Conical Singularities, and Nonuniqueness (Sigurd B. Angenent and Dan Knopf, 2019)4Angenent 2019 Bounded Ricci Curvature and Positive Scalar Curvature under Singular Ricci de Turck Flow (Klaus Kroencke, Tobias Marxen and Boris Vertman, 2020)2Kroencke 2020 A Survey on the Ricci flow on Singular Spaces (Klaus Kroencke and Boris Vertman, 2021)1Kroencke 2021 Uniqueness and stability of singular Ricci flows in higher dimensions (Robert Haslhofer, 2021)8Haslhofer 2021 Perelman's functionals on manifolds with non-isolated conical singularities (Xianzhe Dai and Changliang Wang, 2022)7Dai 2022 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)6Chodosh 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A Survey on the Ricci flow on Singular Spaces

    Klaus Kroencke and Boris Vertman

    Topics Harmonic maps & DeTurck trick, Weak & singular flows, Singularities · Same authors Klaus Kroencke, Boris Vertman · Shared terms “conical singularities”, “isolated conical”, “Turck flow”

  2. Bounded Ricci Curvature and Positive Scalar Curvature under Singular Ricci de Turck Flow

    Klaus Kroencke, Tobias Marxen and Boris Vertman

    Topics Harmonic maps & DeTurck trick, Weak & singular flows, Singularities · Same authors Klaus Kroencke, Boris Vertman · Shared terms “Turck flow”, “conical singularities”, “isolated conical”

  3. Ricci de Turck flow on singular manifolds

    Boris Vertman

    Topics Harmonic maps & DeTurck trick, Singularities · Same author Boris Vertman · Shared terms “Turck flow”

  4. Ricci Solitons, Conical Singularities, and Nonuniqueness

    Sigurd B. Angenent and Dan Knopf

    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”, “expanding Ricci”

  6. Mean Curvature Flow from Conical Singularities

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

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

  7. Perelman's functionals on manifolds with non-isolated conical singularities

    Xianzhe Dai and Changliang Wang

    Topics Singularities · Shared terms “conical singularities”, “isolated conical”, “Perelman”

  8. Uniqueness and stability of singular Ricci flows in higher dimensions

    Robert Haslhofer

    Topics Harmonic maps & DeTurck trick, Weak & singular flows, Singularities

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.