Paper · Mar 10, 2016

Evolution by mean curvature flow of Lagrangian spherical surfaces in complex Euclidean plane

Ildefonso Castro, Ana M. Lerma, Vicente Miquel

math.DGWider flowsarXiv:1603.03229v2
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Abstract

We describe the evolution under the mean curvature flow of embedded Lagrangian spherical surfaces in the complex Euclidean plane . In particular, we answer the Question 4.7 addressed in [Ne10b] by A. Neves about finding out a condition on a starting Lagrangian torus in such that the corresponding mean curvature flow becomes extinct at finite time and converges after rescaling to the Clifford torus.

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Submitted
Mar 10, 2016
Last revised
Feb 9, 2018 (v2)
Subjects
math.DG

Comments: 13 pages. This new version includes drafting changes in some points to make the paper more readable and the deletion of the old Theorem B because the proof was incomplete. Accepted in Journal of Mathematical Analysis and Applications

Mean curvature flowLagrangian mean curvature flowSingularities

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 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 note on the evolution of the Whitney sphere along mean curvature flow (Celso Viana, 2018)3Viana 2018 Ancient solutions in Lagrangian mean curvature flow (Ben Lambert, Jason D. Lotay and Felix Schulze, 2019)5Lambert 2019 Singularities of Equivariant Lagrangian Mean Curvature Flow (Albert Wood, 2019)2Wood 2019 Examples of equivariant Lagrangian mean curvature flow (Jason D. Lotay, 2020)4Lotay 2020 A Priori Estimates for Singularities of the Lagrangian Mean Curvature Flow with Supercritical Phase (Arunima Bhattacharya and Jeremy Wall, 2024)7Bhattacharya 2024 Lagrangian mean curvature flow of surfaces with mean curvature bound (Sourav Ghosh, 2025)1Ghosh 2025 The mean curvature blows up at nondegenerate neck pinch singularities of Lagrangian mean curvature flow (Jason D. Lotay and Goncalo Oliveira, 2026)8Lotay 2026 Existence of Lagrangian Mean Curvature Flow Singularities with Bounded Mean Curvature (Maxwell Stolarski and Wei-Bo Su, 2026)6Stolarski 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 of surfaces with mean curvature bound

    Sourav GhoshWider flows

    Topics Lagrangian mean curvature flow, Singularities · Shared terms “rescaling”

  2. Singularities of Equivariant Lagrangian Mean Curvature Flow

    Albert WoodWider flows

    Topics Lagrangian mean curvature flow, Singularities · Shared terms “rescaling”, “finite time”

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

    Celso VianaWider flows

    Topics Lagrangian mean curvature flow, Singularities · Shared terms “finite time”

  4. Examples of equivariant Lagrangian mean curvature flow

    Jason D. LotayWider flows

    Topics Lagrangian mean curvature flow, Singularities · Shared terms “Clifford torus”, “describe”

  5. Ancient solutions in Lagrangian mean curvature flow

    Ben Lambert, Jason D. Lotay and Felix SchulzeWider flows

    Topics Lagrangian mean curvature flow, Singularities · Shared terms “Clifford torus”

  6. Existence of Lagrangian Mean Curvature Flow Singularities with Bounded Mean Curvature

    Maxwell Stolarski and Wei-Bo SuWider flows

    Topics Lagrangian mean curvature flow, Singularities · Shared terms “finite time”

  7. A Priori Estimates for Singularities of the Lagrangian Mean Curvature Flow with Supercritical Phase

    Arunima Bhattacharya and Jeremy WallWider flows

    Topics Lagrangian mean curvature flow, Singularities

  8. 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

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)

In the guidebook

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  2. 6C.8
    Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.

  3. 11B.4
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    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.