Paper · Jan 22, 2021

A Survey on the Ricci flow on Singular Spaces

Klaus Kroencke, Boris Vertman

math.DGarXiv:2101.09135
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Abstract

In this survey we provide an overview of our recent results concerning Ricci de Turck flow on spaces with isolated conical singularities. The crucial characteristic of the flow is that it preserves the conical singularity. Under certain conditions, Ricci flat metrics with isolated conical singularities are stable and positive scalar curvature is preserved under the flow. We also discuss the relation to Perelman's entropies in the singular setting, and outline open questions and future reseach directions.

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Submitted
Jan 22, 2021
Subjects
math.DG, math.AP
Journal
London Math. Soc. Lecture Notes Series 463 "Differential Geometry in the Large" (2021)
DOI
10.1017/9781108884136

Comments: 17 pages, 1 figure. arXiv admin note: text overlap with arXiv:1802.02908, arXiv:1902.02097, arXiv:2002.06586

Ricci flowRicci-flat & Calabi–YauSingularitiesScalar curvatureWeak & 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 Stability of ALE Ricci-flat manifolds under Ricci flow (Alix Deruelle and Klaus Kroencke, 2017)7Deruelle 2017 Perelman's Entropies for Manifolds with conical Singularities (Klaus Kroencke and Boris Vertman, 2019)1Kroencke 2019 Bounded Ricci Curvature and Positive Scalar Curvature under Singular Ricci de Turck Flow (Klaus Kroencke, Tobias Marxen and Boris Vertman, 2020)2Kroencke 2020 Long-time existence of Yamabe flow on singular spaces with positive Yamabe constant (Jørgen Olsen Lye and Boris Vertman, 2020)5Lye 2020 Convergence of the Ricci flow to Ricci-flat ALE manifolds and positive scalar curvature rigidity (Klaus Kroencke and Oliver Lindblad Petersen, 2020)4Kroencke 2020 Perelman's functionals on manifolds with non-isolated conical singularities (Xianzhe Dai and Changliang Wang, 2022)6Dai 2022 Ricci flow from spaces with edge type conical singularities (Lucas Lavoyer, 2023)8Lavoyer 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Perelman's Entropies for Manifolds with conical Singularities

    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, Scalar curvature · Same authors Klaus Kroencke, Boris Vertman · Shared terms “Turck flow”, “isolated conical”, “conical singularities”

  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. Convergence of the Ricci flow to Ricci-flat ALE manifolds and positive scalar curvature rigidity

    Klaus Kroencke and Oliver Lindblad Petersen

    Topics Ricci-flat & Calabi–Yau, Scalar curvature · Same author Klaus Kroencke · Shared terms “positive scalar”

  5. Long-time existence of Yamabe flow on singular spaces with positive Yamabe constant

    Jørgen Olsen Lye and Boris VertmanWider flows

    Topics Weak & singular flows, Scalar curvature, Singularities · Same author Boris Vertman · Shared terms “singular spaces”

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

    Xianzhe Dai and Changliang Wang

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

  7. Stability of ALE Ricci-flat manifolds under Ricci flow

    Alix Deruelle and Klaus Kroencke

    Topics Ricci-flat & Calabi–Yau · Same author Klaus Kroencke

  8. Ricci flow from spaces with edge type conical singularities

    Lucas Lavoyer

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

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-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

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.