Paper · Feb 12, 2026

Modified mean curvature flow of graphs in Riemannian manifolds

Jocel Faustino Norberto de Oliveira, Jorge Herbert Soares de Lira, Matheus Nunes Soares

math.DGWider flowsarXiv:2602.11930
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Abstract

We obtain height, gradient, and curvature a priori estimates for a modified mean curvature flow in Riemannian manifolds endowed with a Killing vector field. As a consequence, we prove the existence of smooth, entire, longtime solutions for this extrinsic flow with smooth initial data.

Comments: 37 pages. Comments and suggestions are welcome

Mean curvature flowLong-time existence & convergence

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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 A local estimate for the mean curvature flow (Zhen Wang, 2021)8Wang 2021 Graphical mean curvature flow with bounded bi-Ricci curvature (Renan Assimos, Andreas Savas-Halilaj and Knut Smoczyk, 2022)5Assimos 2022 Modified mean curvature flow and CMC foliation conjecture in almost Fuchsian manifolds (Zheng Huang, Longzhi Lin and Zhou Zhang, 2023)3Huang 2023 Foliations of asymptotically Schwarzschildean lightcones by surfaces of constant spacetime mean curvature (Klaus Kroencke and Markus Wolff, 2024)6Kroencke 2024 Dynamical stability of various convex graphical translators (Junyoung Park, 2026)7Park 2026 Mean curvature flow of graphs with asymptotic Dirichlet conditions in Cartan-Hadamard manifolds (Claudia Fernandes, Jorge de Lira and Matheus Soares, 2026)1Fernandes 2026 Translating solitons to higher order mean curvature flows in Riemannian products (Jorge Herbert Soares de Lira and Rafael Rocha de Farias, 2026)4de Lira 2026 Modified Mean Curvature Flow in Fuchsian Manifolds (Yuk Shing Lam and Longzhi Lin, 2026)2Lam 2026
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 of graphs with asymptotic Dirichlet conditions in Cartan-Hadamard manifolds

    Claudia Fernandes, Jorge de Lira and Matheus SoaresWider flows PDF

    Topics Mean curvature flow · Same authors Jorge Herbert Soares de Lira, Matheus Nunes Soares · Shared terms “priori estimates”, “Killing”

  2. Modified Mean Curvature Flow in Fuchsian Manifolds

    Yuk Shing Lam and Longzhi LinWider flows PDF

    Topics Long-time existence & convergence · Shared terms “modified mean”

  3. Modified mean curvature flow and CMC foliation conjecture in almost Fuchsian manifolds

    Zheng Huang, Longzhi Lin and Zhou Zhang PDF

    Topics Long-time existence & convergence · Shared terms “modified mean”

  4. Translating solitons to higher order mean curvature flows in Riemannian products

    Jorge Herbert Soares de Lira and Rafael Rocha de FariasWider flows PDF

    Topics Mean curvature flow · Same author Jorge Herbert Soares de Lira · Shared terms “manifold endowed”

  5. Graphical mean curvature flow with bounded bi-Ricci curvature

    Renan Assimos, Andreas Savas-Halilaj and Knut SmoczykWider flows PDF

    Topics Long-time existence & convergence · Shared terms “Riemannian manifolds”

  6. Foliations of asymptotically Schwarzschildean lightcones by surfaces of constant spacetime mean curvature

    Klaus Kroencke and Markus WolffWider flows PDF

    Topics Long-time existence & convergence · Shared terms “initial data”

  7. Dynamical stability of various convex graphical translators

    Junyoung ParkWider flows PDF

    Topics Long-time existence & convergence · Shared terms “longtime solution”

  8. A local estimate for the mean curvature flow

    Zhen WangWider flows PDF

    Topics Long-time existence & convergence

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.

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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

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.