Paper · Nov 12, 2019

Lagrangian mean curvature flow with boundary

Christopher G. Evans, Ben Lambert, Albert Wood

math.DGWider flowsarXiv:1911.04977
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Abstract

We introduce Lagrangian mean curvature flow with boundary in Calabi–Yau manifolds by defining a natural mixed Dirichlet-Neumann boundary condition, and prove that under this flow, the Lagrangian condition is preserved. We also study in detail the flow of equivariant Lagrangian discs with boundary on the Lawlor neck and the self-shrinking Clifford torus, and demonstrate long-time existence and convergence of the flow in the first instance and of the rescaled flow in the second.

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Submitted
Nov 12, 2019
Subjects
math.DG
Journal
Calc. Var. 61, 106 (2022)
DOI
10.1007/s00526-022-02229-0
Mean curvature flowLagrangian mean curvature flowRicci-flat & Calabi–YauSelf-shrinkers & translatorsKähler geometryBoundaries & non-compactFree boundary & capillaryLong-time existence & convergence

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 2016, 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 A remark on gradient estimates for spacelike mean curvature flow with boundary conditions (Ben Lambert, 2016)5Lambert 2016 Local non-collapsing of volume for the Lagrangian mean curvature flow (Knut Smoczyk, 2018)7Smoczyk 2018 Remarks on the self-shrinking Clifford torus (Christopher G. Evans, Jason D. Lotay and Felix Schulze, 2018)3Evans 2018 Ancient solutions in Lagrangian mean curvature flow (Ben Lambert, Jason D. Lotay and Felix Schulze, 2019)6Lambert 2019 Singularities of Equivariant Lagrangian Mean Curvature Flow (Albert Wood, 2019)1Wood 2019 Examples of equivariant Lagrangian mean curvature flow (Jason D. Lotay, 2020)2Lotay 2020 Lagrangian mean curvature flow in the complex projective plane (Christopher G. Evans, 2022)4Evans 2022 Infinite-Time Singularities of the Lagrangian Mean Curvature Flow (Wei-Bo Su, Chung-Jun Tsai and Albert Wood, 2024)8Su 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Singularities of Equivariant Lagrangian Mean Curvature Flow

    Albert WoodWider flows

    Topics Lagrangian mean curvature flow · Same author Albert Wood · Shared terms “equivariant Lagrangian”, “Lawlor necks”

  2. Examples of equivariant Lagrangian mean curvature flow

    Jason D. LotayWider flows

    Topics Lagrangian mean curvature flow, Long-time existence & convergence, Boundaries & non-compact · Shared terms “equivariant Lagrangian”, “Lawlor necks”, “Clifford torus”

  3. Remarks on the self-shrinking Clifford torus

    Christopher G. Evans, Jason D. Lotay and Felix SchulzeWider flows

    Topics Lagrangian mean curvature flow, Self-shrinkers & translators · Same author Christopher G. Evans · Shared terms “shrinking Clifford”, “Clifford torus”, “self shrinking”

  4. Lagrangian mean curvature flow in the complex projective plane

    Christopher G. EvansWider flows

    Topics Lagrangian mean curvature flow, Ricci-flat & Calabi–Yau, Long-time existence & convergence · Same author Christopher G. Evans · Shared terms “Clifford torus”

  5. A remark on gradient estimates for spacelike mean curvature flow with boundary conditions

    Ben LambertWider flows

    Topics Free boundary & capillary, Self-shrinkers & translators, Long-time existence & convergence · Same author Ben Lambert · Shared terms “boundary condition”, “Neumann boundary”

  6. Ancient solutions in Lagrangian mean curvature flow

    Ben Lambert, Jason D. Lotay and Felix SchulzeWider flows

    Topics Lagrangian mean curvature flow · Same author Ben Lambert · Shared terms “Clifford torus”, “Yau”

  7. Local non-collapsing of volume for the Lagrangian mean curvature flow

    Knut SmoczykWider flows

    Topics Lagrangian mean curvature flow, Ricci-flat & Calabi–Yau, Self-shrinkers & translators

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

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

    Topics Lagrangian mean curvature flow · Same author Albert Wood · Shared terms “Lawlor necks”, “Yau”

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 in the L² sense. 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) 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. 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.