Paper · Sep 17, 2026

Deforming area-preserving maps between surfaces by mean curvature flow coupled with Ricci flow

Ping-Hung Lee

math.DGarXiv:2609.19724
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Abstract

We study a natural way to deform area-preserving maps between compact Riemann surfaces. Specifically, we evolve the metrics on the two Riemann surfaces by the normalized Ricci flow and the graph of the area-preserving map by the mean curvature flow. We prove that the flow exists for all time, remains the graph of an area-preserving map, and converges smoothly and exponentially to a minimal Lagrangian graph with respect to the product of the limiting metrics. This generalizes earlier results of Wang and Smoczyk, in which the Riemann surfaces have constant scalar curvature.

Comments: 27 pages

Ricci flowMean curvature flowLagrangian mean curvature flowSurfacesScalar curvatureLong-time existence & convergenceDiscrete & graph Ricci

Its neighbourhood

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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 Smoothing a measure on a Riemann surface using Ricci flow (Peter M. Topping and Hao Yin, 2021)6Topping 2021 Uniqueness of Ricci flows from nonatomic Radon measures on Riemann surfaces (Peter M. Topping and Hao Yin, 2023)3Topping 2023 Determinants of pseudo-laplacians and ζ^((rm reg))(1) for spinor bundles over Riemann surfaces (Alexey Kokotov and Dmitrii Korikov, 2024)4Kokotov 2024 Dynamical Stability of Minimal Lagrangians in Kähler-Einstein Manifolds of Non-Positive Curvature (Ping-Hung Lee and Chung-Jun Tsai, 2024)1Lee 2024 Branched α-combinatorial Ricci flows on closed surfaces with Euler characteristic χ≤ 0 (Wenjun Li, Rongyuan Liu, Guohao Chen et al., 2025)7Li 2025 The Ricci flow with prescribed curvature on graphs (Yong Lin and Shuang Liu, 2026)5Lin 2026 Infinite-Time Singularities with Vanishing Mean Curvature for Lagrangian Mean Curvature Flow in Gibbons–Hawking Spaces (Ping-Hung Lee and Chung-Jun Tsai, 2026)2Lee 2026 Lagrangian mean curvature flow in the Kummer K3 surface (Carlos Alberto Ochoa Flores, 2026)8Flores 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Dynamical Stability of Minimal Lagrangians in Kähler-Einstein Manifolds of Non-Positive Curvature

    Ping-Hung Lee and Chung-Jun TsaiWider flows PDF

    Topics Lagrangian mean curvature flow, Scalar curvature · Same author Ping-Hung Lee · Shared terms “minimal Lagrangian”, “flow exists”, “Specifically”

  2. Infinite-Time Singularities with Vanishing Mean Curvature for Lagrangian Mean Curvature Flow in Gibbons–Hawking Spaces

    Ping-Hung Lee and Chung-Jun TsaiWider flows PDF

    Topics Lagrangian mean curvature flow · Same author Ping-Hung Lee · Shared terms “flow exists”

  3. Uniqueness of Ricci flows from nonatomic Radon measures on Riemann surfaces

    Peter M. Topping and Hao Yin PDF

    Topics Surfaces · Shared terms “Riemann surface”

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

    Alexey Kokotov and Dmitrii Korikov PDF

    Topics Surfaces · Shared terms “Riemann surface”, “compact Riemann”

  5. The Ricci flow with prescribed curvature on graphs

    Yong Lin and Shuang Liu PDF

    Topics Discrete & graph Ricci, Surfaces, Long-time existence & convergence

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

    Peter M. Topping and Hao Yin PDF

    Topics Surfaces · Shared terms “Riemann surface”

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

    Wenjun Li, Rongyuan Liu, Guohao Chen et al. PDF

    Topics Discrete & graph Ricci, Surfaces, Long-time existence & convergence

  8. Lagrangian mean curvature flow in the Kummer K3 surface

    Carlos Alberto Ochoa FloresWider flows PDF

    Topics Lagrangian mean curvature flow, Long-time existence & convergence · Shared terms “minimal Lagrangian”

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) Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Lagrangian mean curvature flow Wider flows In a Calabi–Yau manifold, mean curvature flow keeps Lagrangian submanifolds Lagrangian. Its hoped-for limits are special Lagrangians, the volume… Start with: A canonical way to deform a Lagrangian submanifold (Smoczyk, 1996) 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. 6D.11
    Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation

    Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.

  3. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.