Paper · Oct 26, 2023

Singularity formation along the line bundle mean curvature flow

Yu Hin Chan, Adam Jacob

math.DGWider flowsarXiv:2310.17709
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Abstract

The line bundle mean curvature flow is a complex analogue of the mean curvature flow for Lagrangian graphs, with fixed points solving the deformed Hermitian-Yang-Mills equation. In this paper we construct two distinct examples of singularities along the flow. First, we find a finite time singularity, ruling out long time existence of the flow in general. Next we show long time existence of the flow with a Calabi symmetry assumption on the blowup of , , if one assumes supercritical phase. Using this, we find an example where a singularity occurs at infinite time along the destabilizing subvariety in the semi-stable case.

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Submitted
Oct 26, 2023
Subjects
math.DG

Comments: 23 pages

Mean curvature flowLagrangian mean curvature flowSingularitiesKähler geometryLong-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.

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A Priori Estimates for Singularities of the Lagrangian Mean Curvature Flow with Supercritical Phase

    Arunima Bhattacharya and Jeremy WallWider flows

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

  2. A convergence result for Mean Curvature Flow of totally real submanifolds

    Tristan C. Collins, Adam Jacob and Yu-Shen LinWider flows

    Topics Lagrangian mean curvature flow · Same author Adam Jacob

  3. Singularity formations in Lagrangian mean curvature flow

    Yang Li and Gábor SzékelyhidiWider flows

    Topics Lagrangian mean curvature flow, Singularities · Shared terms “singularity formation”, “singularities along”

  4. Infinite-Time Singularities with Vanishing Mean Curvature for Lagrangian Mean Curvature Flow in Gibbons–Hawking Spaces

    Ping-Hung Lee and Chung-Jun TsaiWider flows

    Topics Lagrangian mean curvature flow, Singularities · Shared terms “infinite time”, “semi stable”, “time singularity”

  5. Infinite-Time Singularities of the Lagrangian Mean Curvature Flow

    Wei-Bo Su, Chung-Jun Tsai and Albert Wood

    Topics Lagrangian mean curvature flow, Singularities · Shared terms “infinite time”, “singularity formation”, “semi stable”

  6. Special Lagrangian smoothings, Calabi ansatz and stability conditions

    Jacopo StoppaWider flows

    Topics Lagrangian mean curvature flow, Kähler geometry, Singularities · Shared terms “deformed Hermitian”, “Hermitian Yang”, “Yang Mills”

  7. Entire solutions of two-convex Lagrangian mean curvature flows

    Chung-Jun Tsai, Mao-Pei Tsui and Mu-Tao WangWider flows

    Topics Lagrangian mean curvature flow, Long-time existence & convergence

  8. Generic neck pinch singularities along 2D Lagrangian mean curvature flow

    Gábor SzékelyhidiWider flows

    Topics Lagrangian mean curvature flow, Kähler geometry, Singularities · Shared terms “singularities along”, “Calabi”

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) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

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