Paper · Mar 29, 2025

Lagrangian mean curvature flow of surfaces with mean curvature bound

Sourav Ghosh

math.DGWider flowsarXiv:2503.22918
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Abstract

Let be a zero Maslov Lagrangian mean curvature flow in We show that if the mean curvature stays uniformly bounded along the flow, then the tangent flow at a singular point is unique i.e. the limit of the parabolic rescalings does not depend on the chosen sequence of rescalings.

Mean curvature flowLagrangian mean curvature flowSingularities

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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 Neck pinches along the Lagrangian mean curvature flow of surfaces (Jason D. Lotay, Felix Schulze and Gábor Székelyhidi, 2022)3Lotay 2022 A Priori Estimates for Singularities of the Lagrangian Mean Curvature Flow with Supercritical Phase (Arunima Bhattacharya and Jeremy Wall, 2024)5Bhattacharya 2024 Translating Solitons to a Lagrangian mean curvature flow with zero Maslov class (Xiaoli Han, Jiayu Li and Jun Sun, 2024)1Han 2024 Singularity formations in Lagrangian mean curvature flow (Yang Li and Gábor Székelyhidi, 2024)4Li 2024 Type II Singularities of Lagrangian Mean Curvature Flow with Zero Maslov Class (Xiang Li, Yong Luo and Jun Sun, 2024)2Li 2024 Singularities of the Lagrangian mean curvature flow at the critical Lagrangian phase (Arunima Bhattacharya, Ravi Shankar, Jeremy Wall et al., 2025)6Bhattacharya 2025 The mean curvature blows up at nondegenerate neck pinch singularities of Lagrangian mean curvature flow (Jason D. Lotay and Goncalo Oliveira, 2026)8Lotay 2026 Lagrangian mean curvature flow in the Kummer K3 surface (Carlos Alberto Ochoa Flores, 2026)7Flores 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Translating Solitons to a Lagrangian mean curvature flow with zero Maslov class

    Xiaoli Han, Jiayu Li and Jun SunWider flows PDF

    Topics Lagrangian mean curvature flow, Singularities · Shared terms “zero Maslov”

  2. Type II Singularities of Lagrangian Mean Curvature Flow with Zero Maslov Class

    Xiang Li, Yong Luo and Jun SunWider flows PDF

    Topics Lagrangian mean curvature flow, Singularities · Shared terms “zero Maslov”

  3. Neck pinches along the Lagrangian mean curvature flow of surfaces

    Jason D. Lotay, Felix Schulze and Gábor SzékelyhidiWider flows PDF

    Topics Lagrangian mean curvature flow, Singularities · Shared terms “zero Maslov”, “tangent flow”, “Let”

  4. Singularity formations in Lagrangian mean curvature flow

    Yang Li and Gábor SzékelyhidiWider flows PDF

    Topics Lagrangian mean curvature flow, Singularities · Shared terms “tangent flow”

  5. A Priori Estimates for Singularities of the Lagrangian Mean Curvature Flow with Supercritical Phase

    Arunima Bhattacharya and Jeremy WallWider flows PDF

    Topics Lagrangian mean curvature flow, Singularities

  6. Singularities of the Lagrangian mean curvature flow at the critical Lagrangian phase

    Arunima Bhattacharya, Ravi Shankar, Jeremy Wall et al.Wider flows PDF

    Topics Lagrangian mean curvature flow, Singularities

  7. Lagrangian mean curvature flow in the Kummer K3 surface

    Carlos Alberto Ochoa FloresWider flows PDF

    Topics Lagrangian mean curvature flow

  8. The mean curvature blows up at nondegenerate neck pinch singularities of Lagrangian mean curvature flow

    Jason D. Lotay and Goncalo OliveiraWider flows PDF

    Topics Lagrangian mean curvature flow, Singularities

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) 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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993)

In the guidebook

  1. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  2. 6C.8
    Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.

  3. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.