Paper · Dec 13, 2023

Ricci flow of discrete surfaces of revolution, and relation to constant Gaussian curvature

Naoya Suda

math.DGarXiv:2312.08113
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Abstract

Giving explicit parametrizations of discrete constant Gaussian curvature surfaces of revolution that are defined from an integrable systems approach, we study Ricci flow for discrete surfaces, and see how discrete surfaces of revolution have a geometric realization for the Ricci flow that approaches the constant Gaussian curvature surfaces we have parametrized.

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Submitted
Dec 13, 2023
Subjects
math.DG

Comments: 19 pages, 10 figures

Ricci flowRotational symmetry & warped products

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 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 Symmetries of Ricci Flows (Enrique López, Stylianos Dimas and Yuri Bozhkov, 2022)2López 2022 Exponential Asymptotics of Ricci Flow Neckpinch (Hamidreza Mahmoudian, 2023)5Mahmoudian 2023 Gauss Curvature Flow on Surfaces of Revolution – The Noncompact Case (Thalia D. Jeffres and Leonardo Solanilla, 2024)1Jeffres 2024 K^α-translators of offset surfaces (Burcu Bektaş Demirci, Ferdağ Kahraman Aksoyak and Murat Babaarslan, 2024)6Demirci 2024 Geometric properties of second Ricci solitons (Masoumeh Khalili, Ghodratallah Fasihi-Ramandi and Shahroud Azami, 2025)8Khalili 2025 Gradient estimates for a parabolic partial differential equation under the Ricci-Bourguignon flow (José N. V. Gomes, Willian I. Tokura and Hikaru Yamamoto, 2026)7Gomes 2026 A Loewner-Nirenberg phenomena for Ricci flow on compact manifolds with boundary.II (Gang Li, 2026)3Li 2026 A Loewner-Nirenberg phenomena for Ricci flow on compact manifolds with boundary (Gang Li, 2026)4Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Gauss Curvature Flow on Surfaces of Revolution – The Noncompact Case

    Thalia D. Jeffres and Leonardo SolanillaWider flows

    Topics Rotational symmetry & warped products · Shared terms “revolution”

  2. Symmetries of Ricci Flows

    Enrique López, Stylianos Dimas and Yuri Bozhkov

    Topics Rotational symmetry & warped products

  3. A Loewner-Nirenberg phenomena for Ricci flow on compact manifolds with boundary.II

    Gang Li

    Topics Rotational symmetry & warped products

  4. A Loewner-Nirenberg phenomena for Ricci flow on compact manifolds with boundary

    Gang Li

    Topics Rotational symmetry & warped products

  5. Exponential Asymptotics of Ricci Flow Neckpinch

    Hamidreza Mahmoudian

    Topics Rotational symmetry & warped products

  6. -translators of offset surfaces

    Burcu Bektaş Demirci, Ferdağ Kahraman Aksoyak and Murat BabaarslanWider flows

    Topics Rotational symmetry & warped products · Shared terms “revolution”

  7. Gradient estimates for a parabolic partial differential equation under the Ricci-Bourguignon flow

    José N. V. Gomes, Willian I. Tokura and Hikaru Yamamoto

    Topics Rotational symmetry & warped products

  8. Geometric properties of second Ricci solitons

    Masoumeh Khalili, Ghodratallah Fasihi-Ramandi and Shahroud Azami

    Topics Rotational symmetry & warped products

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) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

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. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.