Paper · Jun 29, 2016

The Ricci flow with metric torsion on closed surfaces

Volker Branding, Klaus Kroencke

math.DGarXiv:1606.09121v2
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Abstract

The famous Uniformization Theorem states that on closed Riemannian surfaces there always exists a metric of constant curvature for the Levi-Cevita connection. In this article we prove that an analogue of the uniformization theorem also holds for connections with metric torsion in the case of non-positive Euler characteristic. Our main tool is an adapted form of the Ricci flow.

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Submitted
Jun 29, 2016
Last revised
Jan 7, 2017 (v2)
Subjects
math.DG, math.AP
DOI
10.1007/s12220-016-9753-4

Comments: 14 pages, published version

Ricci flowSurfaces

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 Uniformization of compact foliated spaces by surfaces of hyperbolic type (Richard Muñiz and Alberto Verjovsky, 2021)4Muñiz 2021 On harmonic and biharmonic maps from gradient Ricci solitons (Volker Branding, 2022)7Branding 2022 A Variant Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic (Franziska Borer, Peter Elbau and Tobias Weth, 2023)5Borer 2023 A Simple Ricci Flow Proof of the Uniformization Theorem (Yucheng Ji, 2024)2Ji 2024 Monotonicity of the Liouville entropy along the Ricci flow on surfaces (Karen Butt, Alena Erchenko, Tristan Humbert et al., 2025)8Butt 2025 Branched α-combinatorial Ricci flows on closed surfaces with Euler characteristic χ≤ 0 (Wenjun Li, Rongyuan Liu, Guohao Chen et al., 2025)3Li 2025 Path Connectivity of Anosov Metrics on Surfaces (Guilherme Brandão Guglielmo and R. Ruggiero, 2026)6Guglielmo 2026 Ricci flow with metric torsion on surfaces of positive Euler characteristic (Shubham Dwivedi, 2026)1Dwivedi 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ricci flow with metric torsion on surfaces of positive Euler characteristic

    Shubham Dwivedi

    Topics Surfaces · Shared terms “metric torsion”, “positive Euler”, “Euler characteristic”

  2. A Simple Ricci Flow Proof of the Uniformization Theorem

    Yucheng Ji

    Topics Surfaces · Shared terms “uniformization theorem”

  3. Branched -combinatorial Ricci flows on closed surfaces with Euler characteristic

    Wenjun Li, Rongyuan Liu, Guohao Chen et al.

    Topics Surfaces · Shared terms “closed surfaces”, “Euler characteristic”

  4. Uniformization of compact foliated spaces by surfaces of hyperbolic type

    Richard Muñiz and Alberto Verjovsky

    Topics Surfaces · Shared terms “uniformization”

  5. A Variant Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic

    Franziska Borer, Peter Elbau and Tobias WethWider flows

    Topics Surfaces · Shared terms “closed surfaces”, “Euler characteristic”, “Riemannian surfaces”

  6. Path Connectivity of Anosov Metrics on Surfaces

    Guilherme Brandão Guglielmo and R. Ruggiero

    Topics Surfaces · Shared terms “closed surfaces”

  7. On harmonic and biharmonic maps from gradient Ricci solitons

    Volker Branding

    Topics Surfaces · Same author Volker Branding

  8. Monotonicity of the Liouville entropy along the Ricci flow on surfaces

    Karen Butt, Alena Erchenko, Tristan Humbert et al.

    Topics Surfaces · Shared terms “closed surfaces”

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) 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. 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.