Paper · Jul 17, 2024

A Liouville type theorem for ancient Lagrangian mean curvature flows

Arunima Bhattacharya, Micah Warren, Daniel Weser

math.DGWider flowsarXiv:2407.12733
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Abstract

We prove a Liouville type result for convex solutions of the Lagrangian mean curvature flow with restricted quadratic growth assumptions at antiquity on the solutions.

Comments: 13 pages

Mean curvature flowLagrangian mean curvature flowAncient solutions

Its neighbourhood

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 Ancient solutions and translators of Lagrangian mean curvature flow (Jason D. Lotay, Felix Schulze and Gábor Székelyhidi, 2022)5Lotay 2022 Entire solutions of two-convex Lagrangian mean curvature flows (Chung-Jun Tsai, Mao-Pei Tsui and Mu-Tao Wang, 2023)8Tsai 2023 Hessian estimates for shrinkers, expanders, translators, and rotators of the Lagrangian Mean Curvature Flow (Arunima Bhattacharya and Jeremy Wall, 2024)6Bhattacharya 2024 A Priori Estimates for Singularities of the Lagrangian Mean Curvature Flow with Supercritical Phase (Arunima Bhattacharya and Jeremy Wall, 2024)3Bhattacharya 2024 Lagrangian mean curvature flow of surfaces with mean curvature bound (Sourav Ghosh, 2025)7Ghosh 2025 Asymptotic behavior at infinity and existence of solutions to the Lagrangian mean curvature flow in ℝⁿ⁺¹_- (Jiguang Bao and Zixiao Liu, 2025)1Bao 2025 Singularities of the Lagrangian mean curvature flow at the critical Lagrangian phase (Arunima Bhattacharya, Ravi Shankar, Jeremy Wall et al., 2025)4Bhattacharya 2025 Optimal Transport and Generalized Lagrangian Mean Curvature Flows on Kim-McCann Metrics (Arunima Bhattacharya, Micah Warren and Daniel Weser, 2026)2Bhattacharya 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Asymptotic behavior at infinity and existence of solutions to the Lagrangian mean curvature flow in

    Jiguang Bao and Zixiao LiuWider flows PDF

    Topics Lagrangian mean curvature flow, Ancient solutions · Shared terms “Liouville type”, “quadratic”

  2. Optimal Transport and Generalized Lagrangian Mean Curvature Flows on Kim-McCann Metrics

    Arunima Bhattacharya, Micah Warren and Daniel WeserWider flows PDF

    Topics Lagrangian mean curvature flow · Same authors Arunima Bhattacharya, Micah Warren, Daniel Weser

  3. 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 · Same author Arunima Bhattacharya

  4. 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 · Same author Arunima Bhattacharya

  5. Ancient solutions and translators of Lagrangian mean curvature flow

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

    Topics Lagrangian mean curvature flow, Ancient solutions

  6. 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 author Arunima Bhattacharya

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

    Sourav GhoshWider flows PDF

    Topics Lagrangian mean curvature flow

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

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

    Topics Lagrangian mean curvature flow

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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002)

In the guidebook

  1. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

  2. 12B.1
    κ-SolutionsBook 12B · κ-Solutions and Surgery

    Ancient, nonnegatively curved, noncollapsed: the models for singularities.

  3. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.