Paper · Jul 30, 2021

Smoothing a measure on a Riemann surface using Ricci flow

Peter M. Topping, Hao Yin

math.DGarXiv:2107.14686v4Ars Inveniendi Analytica (2024), Paper No. 4, 28 pp
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Abstract

We formulate and solve the existence problem for Ricci flow on a Riemann surface with initial data given by a Radon measure as volume measure. The theory leads us to a large class of new examples of nongradient expanding Ricci solitons, including the first example of a nongradient Kaehler Ricci soliton. It also settles the question of whether a smooth flow for positive time that attains smooth initial data in a distance metric sense must be smooth down to the initial time. We disprove this by giving an example of a complete Ricci flow starting with the Euclidean plane that is not the static solution.

Ricci flowRicci solitonsKähler–Ricci solitonsSurfacesKähler geometry

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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 2021, 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 Time zero regularity of Ricci flow (Man-Chun Lee and Peter M. Topping, 2022)3Lee 2022 A new complete two-dimensional shrinking gradient Kähler-Ricci soliton (Richard H. Bamler, Charles Cifarelli, Ronan J. Conlon et al., 2022)8Bamler 2022 A Weil-Petersson Type Metric on the Space of Fano Kaehler-Ricci Solitons (Huai-Dong Cao, Xiaofeng Sun and Yingying Zhang, 2022)4Cao 2022 Uniqueness of Ricci flows from nonatomic Radon measures on Riemann surfaces (Peter M. Topping and Hao Yin, 2023)1Topping 2023 All two-dimensional expanding Ricci solitons (Luke T. Peachey and Peter M. Topping, 2023)2Peachey 2023 Determinants of pseudo-laplacians and ζ^((rm reg))(1) for spinor bundles over Riemann surfaces (Alexey Kokotov and Dmitrii Korikov, 2024)5Kokotov 2024 The Cauchy problem for gradient generalized Ricci solitons on a bundle gerbe (Severin Bunk, Miguel Pino Carmona and C. S. Shahbazi, 2025)6Bunk 2025 A non-Kähler expanding Ricci soliton with a Kähler tangent cone at infinity (Richard H. Bamler, Eric Chen and Ronan J. Conlon, 2026)7Bamler 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Uniqueness of Ricci flows from nonatomic Radon measures on Riemann surfaces

    Peter M. Topping and Hao Yin PDF

    Topics Surfaces · Same authors Peter M. Topping, Hao Yin · Shared terms “Riemann surface”, “Radon measure”, “smoothing”

  2. All two-dimensional expanding Ricci solitons

    Luke T. Peachey and Peter M. Topping PDF

    Topics Surfaces, Ricci solitons · Same author Peter M. Topping · Shared terms “complete Ricci”, “expanding Ricci”, “Radon measure”

  3. Time zero regularity of Ricci flow

    Man-Chun Lee and Peter M. Topping PDF

    Topics Ricci flow · Same author Peter M. Topping · Shared terms “attains smooth”, “initial time”, “smooth initial”

  4. A Weil-Petersson Type Metric on the Space of Fano Kaehler-Ricci Solitons

    Huai-Dong Cao, Xiaofeng Sun and Yingying Zhang PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “Kaehler Ricci”

  5. Determinants of pseudo-laplacians and for spinor bundles over Riemann surfaces

    Alexey Kokotov and Dmitrii Korikov PDF

    Topics Surfaces · Shared terms “Riemann surface”

  6. The Cauchy problem for gradient generalized Ricci solitons on a bundle gerbe

    Severin Bunk, Miguel Pino Carmona and C. S. Shahbazi PDF

    Topics Surfaces, Ricci solitons · Shared terms “Riemann surface”, “initial data”

  7. A non-Kähler expanding Ricci soliton with a Kähler tangent cone at infinity

    Richard H. Bamler, Eric Chen and Ronan J. Conlon PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “expanding Ricci”

  8. A new complete two-dimensional shrinking gradient Kähler-Ricci soliton

    Richard H. Bamler, Charles Cifarelli, Ronan J. Conlon et al. PDF

    Topics Surfaces, Kähler–Ricci solitons, Kähler geometry

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) Kähler–Ricci solitons Kähler metrics whose Ricci form differs from the metric by a Lie derivative along a holomorphic vector field. Start with: Uniqueness of Kähler-Ricci solitons (Tian et al., 2000) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.