Paper · Jul 17, 2024

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

Arunima Bhattacharya, Jeremy Wall

math.APWider flowsarXiv:2407.12756v2
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Abstract

In this paper, we prove interior a priori estimates for singularities of the Lagrangian mean curvature flow assuming the Lagrangian phase is supercritical. We prove a Jacobi inequality that holds good when the Lagrangian phase is critical and supercritical. We further extend our results to a broader class of Lagrangian mean curvature type equations.

Comments: arXiv admin note: text overlap with arXiv:2403.07235 Minor revisions were made for clarification. To be published in Nonlinear Analysis

Mean curvature flowLagrangian mean curvature flowSingularities

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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 Singularity formation along the line bundle mean curvature flow (Yu Hin Chan and Adam Jacob, 2023)6Chan 2023 Hessian estimates for shrinkers, expanders, translators, and rotators of the Lagrangian Mean Curvature Flow (Arunima Bhattacharya and Jeremy Wall, 2024)2Bhattacharya 2024 A Liouville type theorem for ancient Lagrangian mean curvature flows (Arunima Bhattacharya, Micah Warren and Daniel Weser, 2024)3Bhattacharya 2024 Singularity formations in Lagrangian mean curvature flow (Yang Li and Gábor Székelyhidi, 2024)7Li 2024 Lagrangian mean curvature flow of surfaces with mean curvature bound (Sourav Ghosh, 2025)4Ghosh 2025 Singularities of the Lagrangian mean curvature flow at the critical Lagrangian phase (Arunima Bhattacharya, Ravi Shankar, Jeremy Wall et al., 2025)1Bhattacharya 2025 Infinite-Time Singularities with Vanishing Mean Curvature for Lagrangian Mean Curvature Flow in Gibbons–Hawking Spaces (Ping-Hung Lee and Chung-Jun Tsai, 2026)8Lee 2026 The mean curvature blows up at nondegenerate neck pinch singularities of Lagrangian mean curvature flow (Jason D. Lotay and Goncalo Oliveira, 2026)5Lotay 2026
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  1. 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 · Same authors Arunima Bhattacharya, Jeremy Wall · Shared terms “Lagrangian phase”, “type equations”, “curvature type”

  2. Hessian estimates for shrinkers, expanders, translators, and rotators of the Lagrangian Mean Curvature Flow

    Arunima Bhattacharya and Jeremy Wall PDF

    Topics Lagrangian mean curvature flow · Same authors Arunima Bhattacharya, Jeremy Wall · Shared terms “Lagrangian phase”, “type equations”, “curvature type”

  3. A Liouville type theorem for ancient Lagrangian mean curvature flows

    Arunima Bhattacharya, Micah Warren and Daniel WeserWider flows PDF

    Topics Lagrangian mean curvature flow · Same author Arunima Bhattacharya

  4. Lagrangian mean curvature flow of surfaces with mean curvature bound

    Sourav GhoshWider flows PDF

    Topics Lagrangian mean curvature flow, Singularities

  5. 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

  6. Singularity formation along the line bundle mean curvature flow

    Yu Hin Chan and Adam JacobWider flows PDF

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

  7. Singularity formations in Lagrangian mean curvature flow

    Yang Li and Gábor SzékelyhidiWider flows PDF

    Topics Lagrangian mean curvature flow, Singularities

  8. 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, 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)

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