Paper · Feb 14, 2022

Lagrangian mean curvature flow in the complex projective plane

Christopher G. Evans

math.DGWider flowsarXiv:2202.06859
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Abstract

We prove a Thomas–Yau-type conjecture for monotone Lagrangian tori satisfying a symmetry condition in the complex projective plane . We show that such tori exist for all time under Lagrangian mean curvature flow with surgery, undergoing at most a finite number of surgeries before flowing to a minimal Clifford torus in infinite time. Furthermore, we show that we can construct a torus with any finite number of surgeries before convergence. Along the way, we prove many interesting subsidiary results and develop methods which should be useful in studying Lagrangian mean curvature flow in non-Calabi–Yau manifolds, even in non-symmetric cases.

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Submitted
Feb 14, 2022
Subjects
math.DG, math.SG

Comments: 35 pages

Mean curvature flowLagrangian mean curvature flowRicci-flat & Calabi–YauSingularitiesSurgeryKähler geometryLong-time existence & convergence

Its neighbourhood

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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 Neck pinches along the Lagrangian mean curvature flow of surfaces (Jason D. Lotay, Felix Schulze and Gábor Székelyhidi, 2022)2Lotay 2022 Neck pinch singularities and Joyce conjectures in Lagrangian mean curvature flow with circle symmetry (Jason D. Lotay and Goncalo Oliveira, 2023)4Lotay 2023 Entire solutions of two-convex Lagrangian mean curvature flows (Chung-Jun Tsai, Mao-Pei Tsui and Mu-Tao Wang, 2023)8Tsai 2023 Lagrangian mean curvature flow of surfaces with mean curvature bound (Sourav Ghosh, 2025)6Ghosh 2025 Generic neck pinch singularities along 2D Lagrangian mean curvature flow (Gábor Székelyhidi, 2026)3Székelyhidi 2026 Optimal Transport and Generalized Lagrangian Mean Curvature Flows on Kim-McCann Metrics (Arunima Bhattacharya, Micah Warren and Daniel Weser, 2026)5Bhattacharya 2026 Infinite-Time Singularities with Vanishing Mean Curvature for Lagrangian Mean Curvature Flow in Gibbons–Hawking Spaces (Ping-Hung Lee and Chung-Jun Tsai, 2026)7Lee 2026 Lagrangian mean curvature flow in the Kummer K3 surface (Carlos Alberto Ochoa Flores, 2026)1Flores 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Lagrangian mean curvature flow in the Kummer K3 surface

    Carlos Alberto Ochoa FloresWider flows

    Topics Lagrangian mean curvature flow, Ricci-flat & Calabi–Yau, Long-time existence & convergence

  2. Neck pinches along the Lagrangian mean curvature flow of surfaces

    Jason D. Lotay, Felix Schulze and Gábor SzékelyhidiWider flows

    Topics Lagrangian mean curvature flow, Ricci-flat & Calabi–Yau, Kähler geometry · Shared terms “Thomas”

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

    Gábor SzékelyhidiWider flows

    Topics Lagrangian mean curvature flow, Ricci-flat & Calabi–Yau, Kähler geometry

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

    Jason D. Lotay and Goncalo OliveiraWider flows

    Topics Lagrangian mean curvature flow, Ricci-flat & Calabi–Yau, Kähler geometry

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

    Arunima Bhattacharya, Micah Warren and Daniel WeserWider flows

    Topics Lagrangian mean curvature flow, Ricci-flat & Calabi–Yau, Long-time existence & convergence

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

    Sourav GhoshWider flows

    Topics Lagrangian mean curvature flow, Singularities

  7. 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”

  8. 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

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) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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)

In the guidebook

  1. 12B.5
    The Canonical Neighbourhood TheoremBook 12B · κ-Solutions and Surgery

    Every high-curvature point looks like a neck or a cap.

  2. 12B.7
    The Singular Time: Horns and Strong NecksBook 12B · κ-Solutions and Surgery

    The limit metric at a singular time, the set Ω where curvature stays bounded, ε-horns and their ends, and strong δ-necks deep in a horn, following Perelman II §3 and Morgan–Tian chapter 11.

  3. 12B.12
    Canonical Neighbourhoods After SurgeryBook 12B · κ-Solutions and Surgery

    The strong canonical neighbourhood assumption justified across surgeries: the blow-up argument, the three cases for a limit that meets a surgery, and the choice of δ (Perelman II §5; Morgan–Tian chapter 17).