Paper · Aug 11, 2026

Correspondence between Mean Curvature Flow and Harmonic-Ricci Flow

Tianyin Ren, Chong Song

math.DGarXiv:2608.10541
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Abstract

In this paper, we observe that the (spacelike) mean curvature flow of a submanifold in a (pseudo-)Euclidean space is equivalent to a harmonic-Ricci flow with coupling constant (or ), for the corresponding Gauss map and the induced metric. The solitons of these two flows are also equivalent. As an application, we get a monotonicity formula for the spacelike mean curvature flow.

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Submitted
Aug 11, 2026
Subjects
math.DG
Ricci flowMean curvature flowPhysics

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 Single copy of the Ricci flow (Rashid Alawadhi, 2022)3Alawadhi 2022 A capillary problem for spacelike mean curvature flow in a cone of Minkowski space (Wilhelm Klingenberg, Ben Lambert and Julian Scheuer, 2023)4Klingenberg 2023 Solutions to the Ricci Flow via Einstein Field Equations (Tommaso Morone and Roberto Tateo, 2024)6Morone 2024 High codimension mean curvature flow of spacelike-convex submanifolds with one spacelike codimension (Ben Andrews and Qiyu Zhou, 2026)8Andrews 2026 Notes on harmonic-Ricci flow on surface (Xiang-Zhi Cao, 2026)1Cao 2026 Functional Renormalization Group as a Ricci Flow: An 𝓕-Entropy Perspective on Information Metric Dynamics (Jin Mo Bok and Ki-Seok Kim, 2026)5Bok 2026 Entropy and Non-Collapse in Lorentzian Geometry (Rohit Dhormare, 2026)7Dhormare 2026 Spacelike Brakke flows with boundary in pseudo-Euclidean space (Javier Hervás, 2026)2Hervás 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Notes on harmonic-Ricci flow on surface

    Xiang-Zhi Cao

    Topics Ricci flow · Shared terms “harmonic Ricci”

  2. Spacelike Brakke flows with boundary in pseudo-Euclidean space

    Javier HervásWider flows

    Topics Physics · Shared terms “spacelike”, “Euclidean space”

  3. Single copy of the Ricci flow

    Rashid Alawadhi

    Topics Physics · Shared terms “correspondence”

  4. A capillary problem for spacelike mean curvature flow in a cone of Minkowski space

    Wilhelm Klingenberg, Ben Lambert and Julian ScheuerWider flows

    Topics Physics · Shared terms “spacelike mean”

  5. Functional Renormalization Group as a Ricci Flow: An -Entropy Perspective on Information Metric Dynamics

    Jin Mo Bok and Ki-Seok Kim

    Topics Physics · Shared terms “coupling constant”, “correspondence”

  6. Solutions to the Ricci Flow via Einstein Field Equations

    Tommaso Morone and Roberto Tateo

    Topics Physics · Shared terms “correspondence”

  7. Entropy and Non-Collapse in Lorentzian Geometry

    Rohit Dhormare

    Topics Physics · Shared terms “correspondence”

  8. High codimension mean curvature flow of spacelike-convex submanifolds with one spacelike codimension

    Ben Andrews and Qiyu ZhouWider flows

    Topics Physics · Shared terms “spacelike”, “submanifolds”, “Euclidean space”

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) Physics Ricci flow first appeared in physics, as the leading-order renormalization-group flow of the string sigma model (Friedan, 1980). Start with: Nonlinear Models in 2+ε Dimensions (Friedan, 1980)

In the guidebook

  1. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.

  2. 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.

  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.