Paper · Feb 16, 2020

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

Klaus Kroencke, Tobias Marxen, Boris Vertman

math.DGarXiv:2002.06586v2
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As on arXiv

Abstract

In this paper we consider a Ricci de Turck flow of spaces with isolated conical singularities, which preserves the conical structure along the flow. We establish that a given initial regularity of Ricci curvature is preserved along the flow. Moreover under additional assumptions, positivity of scalar curvature is preserved under such a flow, mirroring the standard property of Ricci flow on compact manifolds. The analytic difficulty is the a priori low regularity of scalar curvature at the conical tip along the flow, so that the maximum principle does not apply. We view this work as a first step toward studying positivity of the curvature operator along the singular Ricci flow.

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Submitted
Feb 16, 2020
Last revised
Jul 9, 2020 (v2)
Subjects
math.DG, math.AP
Journal
Pacific J. Math. 324 (2023) 295-331
DOI
10.2140/pjm.2023.324.295

Comments: 34 pages, argument in Theorem 7.10 corrected

Ricci flowSingularitiesScalar 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 Singular Ricci flows II (Bruce Kleiner and John Lott, 2018)5Kleiner 2018 Perelman's Entropies for Manifolds with conical Singularities (Klaus Kroencke and Boris Vertman, 2019)1Kroencke 2019 Convergence of the Ricci flow to Ricci-flat ALE manifolds and positive scalar curvature rigidity (Klaus Kroencke and Oliver Lindblad Petersen, 2020)8Kroencke 2020 A Survey on the Ricci flow on Singular Spaces (Klaus Kroencke and Boris Vertman, 2021)2Kroencke 2021 Uniqueness and stability of singular Ricci flows in higher dimensions (Robert Haslhofer, 2021)6Haslhofer 2021 Singular Ricci Flows on surfaces with boundary and positive scalar curvature (Jean C. Cortissoz and Juan J. Villamarín, 2023)4Cortissoz 2023 Ricci-DeTurck Flow from Initial Metric with Morrey-type Integrability Condition (Man-Chun Lee and Stephen Shang Yi Liu, 2024)7Lee 2024
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 “Turck flow”, “conical singularities”, “isolated conical”

  2. A Survey on the Ricci flow on Singular Spaces

    Klaus Kroencke 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. Singular Ricci Flows on surfaces with boundary and positive scalar curvature

    Jean C. Cortissoz and Juan J. Villamarín

    Topics Weak & singular flows, Scalar curvature · Shared terms “positive scalar”

  5. Singular Ricci flows II

    Bruce Kleiner and John Lott

    Topics Weak & singular flows

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

    Robert Haslhofer

    Topics Harmonic maps & DeTurck trick, Weak & singular flows, Singularities · Shared terms “maximum principle”

  7. Ricci-DeTurck Flow from Initial Metric with Morrey-type Integrability Condition

    Man-Chun Lee and Stephen Shang Yi Liu

    Topics Harmonic maps & DeTurck trick, Weak & singular flows, Scalar curvature

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

    Klaus Kroencke and Oliver Lindblad Petersen

    Topics Scalar curvature · Same author Klaus Kroencke · Shared terms “positive scalar”

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) 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) 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.