Paper · Sep 18, 2023

Entire solutions of two-convex Lagrangian mean curvature flows

Chung-Jun Tsai, Mao-Pei Tsui, Mu-Tao Wang

math.DGWider flowsarXiv:2309.09432v2
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Abstract

Given an entire function on , we consider the graph of as a Lagrangian submanifold of , and deform it by the mean curvature flow in . This leads to the special Lagrangian evolution equation, a fully nonlinear Hessian type PDE. We prove long-time existence and convergence results under a 2-positivity assumption of . Such results were previously known only under the stronger assumption of positivity of .

Comments: 21 pages, 3 figures. Typos corrected. Minor modifications. To appear in Calc. Var. Partial Differential Equations

Mean curvature flowLagrangian 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 Mean Curvature Flows of Two-Convex Lagrangians (Chung-Jun Tsai, Mao-Pei Tsui and Mu-Tao Wang, 2023)1Tsai 2023 Infinite-Time Singularities of the Lagrangian Mean Curvature Flow (Wei-Bo Su, Chung-Jun Tsai and Albert Wood, 2024)4Su 2024 Stability of the generalized Lagrangian mean curvature flow in cotangent bundle (Xishen Jin and Jiawei Liu, 2024)8Jin 2024 Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space II (Shanshan Li, Jiaru Lv and Rongli Huang, 2024)6Li 2024 Lagrangian mean curvature flow of surfaces with mean curvature bound (Sourav Ghosh, 2025)7Ghosh 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)3Lee 2026 Almost isoclinic Lagrangian submanifolds (Tang-Kai Lee and Mu-Tao Wang, 2026)2Lee 2026 Lagrangian mean curvature flow in the Kummer K3 surface (Carlos Alberto Ochoa Flores, 2026)5Flores 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 Flows of Two-Convex Lagrangians

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

    Topics Lagrangian mean curvature flow, Long-time existence & convergence · Same authors Chung-Jun Tsai, Mao-Pei Tsui, Mu-Tao Wang · Shared terms “convex lagrangian”, “Lagrangian submanifold”, “previously”

  2. Almost isoclinic Lagrangian submanifolds

    Tang-Kai Lee and Mu-Tao WangWider flows PDF

    Topics Lagrangian mean curvature flow · Same author Mu-Tao Wang · Shared terms “Lagrangian submanifold”

  3. 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 · Same author Chung-Jun Tsai · Shared terms “special Lagrangian”

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

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

    Topics Lagrangian mean curvature flow · Same author Chung-Jun Tsai · Shared terms “special Lagrangian”

  5. Lagrangian mean curvature flow in the Kummer K3 surface

    Carlos Alberto Ochoa FloresWider flows PDF

    Topics Lagrangian mean curvature flow, Long-time existence & convergence · Shared terms “special Lagrangian”

  6. Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space II

    Shanshan Li, Jiaru Lv and Rongli HuangWider flows PDF

    Topics Lagrangian mean curvature flow · Shared terms “special Lagrangian”

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

    Sourav GhoshWider flows PDF

    Topics Lagrangian mean curvature flow

  8. Stability of the generalized Lagrangian mean curvature flow in cotangent bundle

    Xishen Jin and Jiawei LiuWider flows PDF

    Topics Lagrangian mean curvature flow · Shared terms “Lagrangian submanifold”, “special Lagrangian”

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) 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.