Paper · Apr 29, 2022

Ancient solutions and translators of Lagrangian mean curvature flow

Jason D. Lotay, Felix Schulze, Gábor Székelyhidi

math.DGWider flowsarXiv:2204.13836v2Publ. math. IHES (2024)
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Abstract

Suppose that is an almost calibrated, exact, ancient solution of Lagrangian mean curvature flow in . We show that if has a blow-down given by the static union of two Lagrangian subspaces with distinct Lagrangian angles that intersect along a line, then is a translator. In particular in , all almost calibrated, exact, ancient solutions of Lagrangian mean curvature flow with entropy less than 3 are special Lagrangian, a union of planes, or translators.

Comments: 28 pages, v2: appendix added on smoothly immersed Brakke flows, accepted in Publ. Math. IHES

Mean curvature flowLagrangian mean curvature flowAncient solutionsSelf-shrinkers & translators

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 Neck pinches along the Lagrangian mean curvature flow of surfaces (Jason D. Lotay, Felix Schulze and Gábor Székelyhidi, 2022)1Lotay 2022 Neck pinch singularities and Joyce conjectures in Lagrangian mean curvature flow with circle symmetry (Jason D. Lotay and Goncalo Oliveira, 2023)3Lotay 2023 Infinite-Time Singularities of the Lagrangian Mean Curvature Flow (Wei-Bo Su, Chung-Jun Tsai and Albert Wood, 2024)8Su 2024 A Liouville type theorem for ancient Lagrangian mean curvature flows (Arunima Bhattacharya, Micah Warren and Daniel Weser, 2024)5Bhattacharya 2024 Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space II (Shanshan Li, Jiaru Lv and Rongli Huang, 2024)7Li 2024 Singularity formations in Lagrangian mean curvature flow (Yang Li and Gábor Székelyhidi, 2024)2Li 2024 Type II Singularities of Lagrangian Mean Curvature Flow with Zero Maslov Class (Xiang Li, Yong Luo and Jun Sun, 2024)6Li 2024 The mean curvature blows up at nondegenerate neck pinch singularities of Lagrangian mean curvature flow (Jason D. Lotay and Goncalo Oliveira, 2026)4Lotay 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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 · Same authors Jason D. Lotay, Felix Schulze, Gábor Székelyhidi · Shared terms “static union”, “Suppose”

  2. Singularity formations in Lagrangian mean curvature flow

    Yang Li and Gábor SzékelyhidiWider flows PDF

    Topics Lagrangian mean curvature flow · Same author Gábor Székelyhidi · Shared terms “special Lagrangian”

  3. Neck pinch singularities and Joyce conjectures in Lagrangian mean curvature flow with circle symmetry

    Jason D. Lotay and Goncalo OliveiraWider flows PDF

    Topics Lagrangian mean curvature flow · Same author Jason D. Lotay · Shared terms “almost calibrated”, “special Lagrangian”

  4. 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 · Same author Jason D. Lotay

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

    Arunima Bhattacharya, Micah Warren and Daniel WeserWider flows PDF

    Topics Lagrangian mean curvature flow, Ancient solutions

  6. 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, Self-shrinkers & translators · Shared terms “almost calibrated”

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

    Shanshan Li, Jiaru Lv and Rongli HuangWider flows PDF

    Topics Lagrangian mean curvature flow, Self-shrinkers & translators · Shared terms “special Lagrangian”

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

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

    Topics Lagrangian mean curvature flow, Ancient solutions · Shared terms “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) 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) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990)

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.