Paper · Apr 25, 2019

Loss of initial data under limits of Ricci flows

Peter M. Topping

math.DGarXiv:1904.11232v2
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Abstract

We construct a sequence of smooth Ricci flows on , with standard uniform curvature decay, and with initial metrics converging to the standard flat unit-area square torus in the Gromov-Hausdorff sense, with the property that the flows themselves converge not to the static Ricci flow , but to the static Ricci flow of twice the area.

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Submitted
Apr 25, 2019
Last revised
Sep 1, 2021 (v2)
Subjects
math.DG
Journal
`Minimal surfaces: integrable systems and visualisation,' [Granada, 2018] T. Hoffmann, M. Kilian, K. Leschke, F. Martin (Eds.) Springer Proceedings in Mathematics & Statistics 349, 2021
DOI
10.1007/978-3-030-68541-6

Comments: 4 pages. This is the submitted version from 2019, but with updated references. Problem 1.2 can be answered using the theory in [19]

Ricci flowLimit 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 Rate of curvature decay for the contracting cusp Ricci flow (Peter M. Topping and Hao Yin, 2016)5Topping 2016 Local mollification of Riemannian metrics using Ricci flow, and Ricci limit spaces (Miles Simon and Peter M. Topping, 2017)4Simon 2017 Global Regularity of Three-Dimensional Ricci Limit Spaces (Andrew D. McLeod and Peter M. Topping, 2018)2McLeod 2018 Ricci flow and Ricci Limit Spaces (Peter M. Topping, 2019)3Topping 2019 Rigidity of the first Betti number via Ricci flow smoothing (Shaosai Huang and Bing Wang, 2020)7Huang 2020 Smoothing a measure on a Riemann surface using Ricci flow (Peter M. Topping and Hao Yin, 2021)6Topping 2021 Time zero regularity of Ricci flow (Man-Chun Lee and Peter M. Topping, 2022)1Lee 2022 Geometric regularity of blow-up limits of the Kähler-Ricci flow (Max Hallgren, Wangjian Jian, Jian Song et al., 2023)8Hallgren 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Time zero regularity of Ricci flow

    Man-Chun Lee and Peter M. Topping

    Topics Limit spaces & RCD · Same author Peter M. Topping · Shared terms “smooth Ricci”, “initial data”, “Gromov Hausdorff”

  2. Global Regularity of Three-Dimensional Ricci Limit Spaces

    Andrew D. McLeod and Peter M. Topping

    Topics Limit spaces & RCD · Same author Peter M. Topping

  3. Ricci flow and Ricci Limit Spaces

    Peter M. Topping

    Topics Limit spaces & RCD · Same author Peter M. Topping

  4. Local mollification of Riemannian metrics using Ricci flow, and Ricci limit spaces

    Miles Simon and Peter M. Topping

    Topics Limit spaces & RCD · Same author Peter M. Topping

  5. Rate of curvature decay for the contracting cusp Ricci flow

    Peter M. Topping and Hao Yin

    Topics Ricci flow · Same author Peter M. Topping · Shared terms “curvature decay”

  6. Smoothing a measure on a Riemann surface using Ricci flow

    Peter M. Topping and Hao Yin

    Topics Ricci flow · Same author Peter M. Topping · Shared terms “initial data”

  7. Rigidity of the first Betti number via Ricci flow smoothing

    Shaosai Huang and Bing Wang

    Topics Limit spaces & RCD · Shared terms “initial data”, “Gromov Hausdorff”

  8. Geometric regularity of blow-up limits of the Kähler-Ricci flow

    Max Hallgren, Wangjian Jian, Jian Song et al.

    Topics Limit spaces & RCD · Shared terms “Gromov Hausdorff”, “sequence”

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) 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. 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.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.