Paper · Feb 6, 2018

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

Celso Viana

math.DGWider flowsarXiv:1802.02108v2
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Abstract

We study the evolution of the Whitney sphere along the Lagrangian mean curvature flow. We show that equivariant Lagrangian spheres in satisfying mild geometric assumptions collapse to a point in finite time and the tangent flows converge to a Lagrangian plane with multiplicity two.

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Submitted
Feb 6, 2018
Last revised
Dec 12, 2018 (v2)
Subjects
math.DG

Comments: 13 pages, typos corrected, 2 figures

Mean curvature flowLagrangian mean curvature flowSingularities

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 Evolution by mean curvature flow of Lagrangian spherical surfaces in complex Euclidean plane (Ildefonso Castro, Ana M. Lerma and Vicente Miquel, 2016)6Castro 2016 Lagrangian mean curvature flow of Whitney spheres (Andreas Savas-Halilaj and Knut Smoczyk, 2018)1Savas-Halila 2018 Singularities of Equivariant Lagrangian Mean Curvature Flow (Albert Wood, 2019)3Wood 2019 Examples of equivariant Lagrangian mean curvature flow (Jason D. Lotay, 2020)2Lotay 2020 Singularity formations in Lagrangian mean curvature flow (Yang Li and Gábor Székelyhidi, 2024)5Li 2024 Lagrangian mean curvature flow of surfaces with mean curvature bound (Sourav Ghosh, 2025)4Ghosh 2025 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 Existence of Lagrangian Mean Curvature Flow Singularities with Bounded Mean Curvature (Maxwell Stolarski and Wei-Bo Su, 2026)8Stolarski 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 Whitney spheres

    Andreas Savas-Halilaj and Knut SmoczykWider flows

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

  2. Examples of equivariant Lagrangian mean curvature flow

    Jason D. LotayWider flows

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

  3. Singularities of Equivariant Lagrangian Mean Curvature Flow

    Albert WoodWider flows

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

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

    Sourav GhoshWider flows

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

  5. Singularity formations in Lagrangian mean curvature flow

    Yang Li and Gábor SzékelyhidiWider flows

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

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

    Ildefonso Castro, Ana M. Lerma and Vicente MiquelWider flows

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

  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 “Lagrangian spheres”

  8. 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”

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
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

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