Paper · Oct 31, 2016

Ricci flow from spaces with isolated conical singularities

Panagiotis Gianniotis, Felix Schulze

math.DGarXiv:1610.09753v2
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Abstract

Let be a compact -dimensional Riemannian manifold with a finite number of singular points, where the metric is asymptotic to a non-negatively curved cone over . We show that there exists a smooth Ricci flow starting from such a metric with curvature decaying like C/t. The initial metric is attained in Gromov-Hausdorff distance and smoothly away from the singular points. In the case that the initial manifold has isolated singularities asymptotic to a non-negatively curved cone over , where acts freely and properly discontinuously, we extend the above result by showing that starting from such an initial condition there exists a smooth Ricci flow with isolated orbifold singularities.

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Submitted
Oct 31, 2016
Last revised
Jul 9, 2018 (v2)
Subjects
math.DG
Journal
Geom. Topol. 22 (2018) 3925-3977
DOI
10.2140/gt.2018.22.3925

Comments: Final version, to appear in Geometry & Topology

Ricci flowSingularitiesLimit spaces & RCD

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 Uniqueness of asymptotically conical tangent flows (Otis Chodosh and Felix Schulze, 2019)8Chodosh 2019 A gradient flow of isometric G₂ structures (Shubham Dwivedi, Panagiotis Gianniotis and Spiro Karigiannis, 2019)5Dwivedi 2019 On the regularity of Ricci flows coming out of metric spaces (Alix Deruelle, Felix Schulze and Miles Simon, 2019)6Deruelle 2019 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)1Lavoyer 2023 Mean Curvature Flow from Conical Singularities (Otis Chodosh, J. M. Daniels-Holgate and Felix Schulze, 2023)2Chodosh 2023 Splitting maps in Type I Ricci flows (Panagiotis Gianniotis, 2024)4Gianniotis 2024 Diameter bounds in 3d Type I Ricci flows (Panagiotis Gianniotis, 2025)3Gianniotis 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ricci flow from spaces with edge type conical singularities

    Lucas Lavoyer

    Topics Limit spaces & RCD, Singularities · Shared terms “conical singularities”, “curved cone”, “Gromov Hausdorff”

  2. Mean Curvature Flow from Conical Singularities

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

    Topics Singularities · Same author Felix Schulze · Shared terms “conical singularities”, “isolated conical”

  3. Diameter bounds in 3d Type I Ricci flows

    Panagiotis Gianniotis

    Topics Limit spaces & RCD, Singularities · Same author Panagiotis Gianniotis

  4. Splitting maps in Type I Ricci flows

    Panagiotis Gianniotis

    Topics Limit spaces & RCD, Singularities · Same author Panagiotis Gianniotis

  5. A gradient flow of isometric structures

    Shubham Dwivedi, Panagiotis Gianniotis and Spiro Karigiannis

    Topics Limit spaces & RCD, Singularities · Same author Panagiotis Gianniotis · Shared terms “Hausdorff”, “Gromov”

  6. On the regularity of Ricci flows coming out of metric spaces

    Alix Deruelle, Felix Schulze and Miles Simon

    Topics Limit spaces & RCD · Same author Felix Schulze · Shared terms “Gromov Hausdorff”

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

    Xianzhe Dai and Changliang Wang

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

  8. Uniqueness of asymptotically conical tangent flows

    Otis Chodosh and Felix SchulzeWider flows

    Topics Singularities · Same author Felix Schulze · Shared terms “conical singularities”, “singular point”, “isolated”

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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

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. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.