Paper · Aug 18, 2020

Examples of equivariant Lagrangian mean curvature flow

Jason D. Lotay

math.DGWider flowsarXiv:2008.08067
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Abstract

In this expository note we describe important examples of Lagrangian mean curvature flow in which are invariant under a circle action. Through these examples, we see compact and non-compact situations, long-time existence, singularities forming via explicit models, and significant objects in Riemannian and symplectic geometry, including the Clifford torus, Chekanov torus, Whitney sphere and Lawlor necks.

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Submitted
Aug 18, 2020
Subjects
math.DG, math.AP, math.SG

Comments: 8 pages, 7 figures. An abridged version will appear in Proceedings of the 2019 ISAAC Congress (Aveiro, Portugal)

Mean curvature flowLagrangian mean curvature flowSingularitiesBoundaries & non-compactLong-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 Remarks on the self-shrinking Clifford torus (Christopher G. Evans, Jason D. Lotay and Felix Schulze, 2018)8Evans 2018 A note on the evolution of the Whitney sphere along mean curvature flow (Celso Viana, 2018)4Viana 2018 Lagrangian mean curvature flow of Whitney spheres (Andreas Savas-Halilaj and Knut Smoczyk, 2018)2Savas-Halila 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 Lagrangian mean curvature flow with boundary (Christopher G. Evans, Ben Lambert and Albert Wood, 2019)3Evans 2019 Neck pinch singularities and Joyce conjectures in Lagrangian mean curvature flow with circle symmetry (Jason D. Lotay and Goncalo Oliveira, 2023)7Lotay 2023 The mean curvature blows up at nondegenerate neck pinch singularities of Lagrangian mean curvature flow (Jason D. Lotay and Goncalo Oliveira, 2026)5Lotay 2026
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, Singularities · Shared terms “equivariant Lagrangian”, “Lawlor necks”

  2. Lagrangian mean curvature flow of Whitney spheres

    Andreas Savas-Halilaj and Knut SmoczykWider flows

    Topics Lagrangian mean curvature flow, Singularities · Shared terms “Whitney sphere”, “equivariant Lagrangian”

  3. Lagrangian mean curvature flow with boundary

    Christopher G. Evans, Ben Lambert and Albert WoodWider flows

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

  4. A note on the evolution of the Whitney sphere along mean curvature flow

    Celso VianaWider flows

    Topics Lagrangian mean curvature flow, Singularities · Shared terms “Whitney sphere”, “equivariant Lagrangian”

  5. The mean curvature blows up at nondegenerate neck pinch singularities of Lagrangian mean curvature flow

    Jason D. Lotay and Goncalo OliveiraWider flows

    Topics Lagrangian mean curvature flow, Singularities · Same author Jason D. Lotay

  6. Ancient solutions in Lagrangian mean curvature flow

    Ben Lambert, Jason D. Lotay and Felix SchulzeWider flows

    Topics Lagrangian mean curvature flow, Singularities · Same author Jason D. Lotay · Shared terms “Clifford torus”

  7. 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, Singularities · Same author Jason D. Lotay

  8. Remarks on the self-shrinking Clifford torus

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

    Topics Lagrangian mean curvature flow · Same author Jason D. Lotay · Shared terms “Clifford torus”

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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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