Paper · Oct 27, 2025

Mean curvature flow into an ambient Riemannian manifold evolving by Ricci flow coupled with harmonic map heat flow

José N. V. Gomes, Matheus Hudson, Carlos M. de Sousa

math.DGarXiv:2510.23239
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Abstract

The main objective of this article is to study the mean curvature flow into an ambient compact smooth manifold M with boundary and with a Riemannian metric that evolves by a self-similar solution of the Ricci flow coupled with the harmonic map heat flow of a map from M to a Riemannian manifold N. In this context, we address a functional associated with this flow and calculate its variation along parameters that preserve the weighted volume measure. An extension of Hamilton's differential Harnack expression appears by considering the boundary of M evolving by mean curvature flow, which must vanish on the gradient steady soliton case. Next, we obtain a Huisken monotonicity-type formula for the mean curvature flow in the proposed background. We also show how to construct a family of mean curvature solitons and establish a characterization of such a family.

Comments: 30 pages

Ricci flowMean curvature flowRicci solitonsHarnack inequalitiesHarmonic maps & DeTurck trick

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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 Short time existence for harmonic map heat flow with time-dependent metrics (Shaochuang Huang and Luen-Fai Tam, 2021)3Huang 2021 Rigidity of Lipschitz map using harmonic map heat flow (Man-Chun Lee and Luen-Fai Tam, 2022)4Lee 2022 Mean curvature flow in an extended Ricci flow background (José N. V. Gomes and Matheus Hudson, 2023)1Gomes 2023 Scalar curvature rigidity of the four-dimensional sphere (Simone Cecchini, Jinmin Wang, Zhizhang Xie et al., 2024)7Cecchini 2024 Gradient estimation of a generalized non-linear heat type equation along Super-Perelman Ricci flow on weighted Riemannian manifolds (Yanlin Li, Abimbola Abolarinwa, Suraj Ghosh et al., 2024)6Li 2024 Log gradient estimates for heat type equations on manifolds (Qi S. Zhang, 2024)8Zhang 2024 A note on Huisken monotonicity-type formula for the mean curvature flow in a gradient shrinking extended Ricci soliton background (José N. V. Gomes, Matheus Hudson and Hikaru Yamamoto, 2024)2Gomes 2024 Convergence of Ricci flow and long-time existence of Harmonic map heat flow (Kyeongsu Choi and Yi Lai, 2025)5Choi 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Mean curvature flow in an extended Ricci flow background

    José N. V. Gomes and Matheus Hudson PDF

    Topics Harnack inequalities, Ricci solitons · Same authors José N. V. Gomes, Matheus Hudson · Shared terms “Harnack expression”, “Hamilton differential”, “monotonicity type”

  2. A note on Huisken monotonicity-type formula for the mean curvature flow in a gradient shrinking extended Ricci soliton background

    José N. V. Gomes, Matheus Hudson and Hikaru Yamamoto PDF

    Topics Ricci solitons · Same authors José N. V. Gomes, Matheus Hudson · Shared terms “monotonicity type”, “Huisken monotonicity”, “type formula”

  3. Short time existence for harmonic map heat flow with time-dependent metrics

    Shaochuang Huang and Luen-Fai Tam PDF

    Topics Harmonic maps & DeTurck trick · Shared terms “map heat”, “heat flow”, “flow coupled”

  4. Rigidity of Lipschitz map using harmonic map heat flow

    Man-Chun Lee and Luen-Fai Tam PDF

    Topics Harmonic maps & DeTurck trick · Shared terms “map heat”, “heat flow”, “flow coupled”

  5. Convergence of Ricci flow and long-time existence of Harmonic map heat flow

    Kyeongsu Choi and Yi Lai PDF

    Topics Harmonic maps & DeTurck trick, Ricci solitons · Shared terms “map heat”, “heat flow”

  6. Gradient estimation of a generalized non-linear heat type equation along Super-Perelman Ricci flow on weighted Riemannian manifolds

    Yanlin Li, Abimbola Abolarinwa, Suraj Ghosh et al. PDF

    Topics Harnack inequalities · Shared terms “manifold evolving”, “Riemannian manifolds”

  7. Scalar curvature rigidity of the four-dimensional sphere

    Simone Cecchini, Jinmin Wang, Zhizhang Xie et al. PDF

    Topics Harmonic maps & DeTurck trick · Shared terms “map heat”, “flow coupled”, “heat flow”

  8. Log gradient estimates for heat type equations on manifolds

    Qi S. Zhang PDF

    Topics Harnack inequalities · Shared terms “Hamilton”

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) 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) Harnack inequalities Differential Harnack inequalities compare a solution with itself at different points and times. Hamilton's matrix Harnack inequality says curvature… Start with: On the parabolic kernel of the Schrödinger operator (Li et al., 1986)

In the guidebook

  1. 6D.5
    Diffusions on Manifolds and CurvatureBook 6D · Diffusions, Finance and Computation

    Brownian motion on a Riemannian manifold, the Bakry–Émery calculus, curvature-dimension inequalities, gradient estimates and log-Sobolev, coupling, and Brownian motion along a Ricci flow.

  2. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

  3. 12A.6
    𝓛-Geodesics and the Variational FormulasBook 12A · Entropy and Noncollapsing

    Perelman's §7.1 derived line by line: the first and second variation of 𝓛-length, 𝓛-Jacobi fields, adapted fields, Hamilton's Harnack expressions, and the differential inequalities for the reduced distance.