Paper · Oct 14, 2019

Singularities of Equivariant Lagrangian Mean Curvature Flow

Albert Wood

math.DGWider flowsarXiv:1910.06122v2
View PDF View HTML arXiv page

As on arXiv

Abstract

We study almost-calibrated, -equivariant Lagrangian mean curvature flow in , and prove structural theorems about the Type I and Type II blowups of finite-time singularities. In particular, we prove that any Type I blowup of such a flow must be a special Lagrangian pair of transversely intersecting planes, any Type II blowup must be the Lawlor neck with the same asymptotes, and these blowups are independent of the choice of rescaling. We also give a partial classification of when singularities occur in the equivariant case, and examine the intermediate scales between the Type I and Type II models.

View PDF View HTML arXiv page
Submitted
Oct 14, 2019
Last revised
Dec 7, 2020 (v2)
Subjects
math.DG, math.AP

Comments: 30 pages, 9 figures. v2: strengthened results slightly (embeddedness no longer a necessary assumption in main theorems), corrected minor errors (in particular replaced Lemma 4.7 of v1), improved exposition in places

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 Lagrangian mean curvature flow of Whitney spheres (Andreas Savas-Halilaj and Knut Smoczyk, 2018)6Savas-Halila 2018 Ancient solutions in Lagrangian mean curvature flow (Ben Lambert, Jason D. Lotay and Felix Schulze, 2019)7Lambert 2019 Lagrangian mean curvature flow with boundary (Christopher G. Evans, Ben Lambert and Albert Wood, 2019)2Evans 2019 Examples of equivariant Lagrangian mean curvature flow (Jason D. Lotay, 2020)1Lotay 2020 Cohomogeneity-One Lagrangian Mean Curvature Flow (Jesse Madnick and Albert Wood, 2022)5Madnick 2022 Infinite-Time Singularities of the Lagrangian Mean Curvature Flow (Wei-Bo Su, Chung-Jun Tsai and Albert Wood, 2024)3Su 2024 Singularity formations in Lagrangian mean curvature flow (Yang Li and Gábor Székelyhidi, 2024)4Li 2024 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. Examples of equivariant Lagrangian mean curvature flow

    Jason D. LotayWider flows

    Topics Lagrangian mean curvature flow, Singularities · Shared terms “equivariant Lagrangian”, “Lawlor necks”

  2. Lagrangian mean curvature flow with boundary

    Christopher G. Evans, Ben Lambert and Albert WoodWider flows

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

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

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

    Topics Lagrangian mean curvature flow, Singularities · Same author Albert Wood · Shared terms “special Lagrangian”, “time singularity”, “Lawlor necks”

  4. Singularity formations in Lagrangian mean curvature flow

    Yang Li and Gábor SzékelyhidiWider flows

    Topics Lagrangian mean curvature flow, Singularities · Shared terms “Lawlor necks”, “special Lagrangian”

  5. Cohomogeneity-One Lagrangian Mean Curvature Flow

    Jesse Madnick and Albert Wood

    Topics Lagrangian mean curvature flow, Singularities · Same author Albert Wood · Shared terms “special Lagrangian”, “almost calibrated”, “time singularity”

  6. Lagrangian mean curvature flow of Whitney spheres

    Andreas Savas-Halilaj and Knut SmoczykWider flows

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

  7. Ancient solutions in Lagrangian mean curvature flow

    Ben Lambert, Jason D. Lotay and Felix SchulzeWider flows

    Topics Lagrangian mean curvature flow, Singularities · Shared terms “almost calibrated”, “structural”

  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”, “time singularity”, “structural”

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

  1. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  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.