Paper · Sep 25, 2026

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

Jason D. Lotay, Goncalo Oliveira

math.DGWider flowsarXiv:2609.31343
PDF Abstract page

As on arXiv

Abstract

We prove that at any nondegenerate neck pinch singularity of Lagrangian mean curvature flow of surfaces the mean curvature becomes unbounded, with control on the blow-up rate.

Comments: 11 pages

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 2021, 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 Ancient solutions and translators of Lagrangian mean curvature flow (Jason D. Lotay, Felix Schulze and Gábor Székelyhidi, 2022)4Lotay 2022 Neck pinches along the Lagrangian mean curvature flow of surfaces (Jason D. Lotay, Felix Schulze and Gábor Székelyhidi, 2022)3Lotay 2022 Neck pinch singularities and Joyce conjectures in Lagrangian mean curvature flow with circle symmetry (Jason D. Lotay and Goncalo Oliveira, 2023)1Lotay 2023 Singularity formations in Lagrangian mean curvature flow (Yang Li and Gábor Székelyhidi, 2024)7Li 2024 Lagrangian mean curvature flow of surfaces with mean curvature bound (Sourav Ghosh, 2025)6Ghosh 2025 Generic neck pinch singularities along 2D Lagrangian mean curvature flow (Gábor Székelyhidi, 2026)2Székelyhidi 2026 Nondegenerate neck pinches along the mean curvature flow (Gábor Székelyhidi, 2026)8Székelyhidi 2026 Infinite-Time Singularities with Vanishing Mean Curvature for Lagrangian Mean Curvature Flow in Gibbons–Hawking Spaces (Ping-Hung Lee and Chung-Jun Tsai, 2026)5Lee 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Neck pinch singularities and Joyce conjectures in Lagrangian mean curvature flow with circle symmetry

    Jason D. Lotay and Goncalo OliveiraWider flows PDF

    Topics Lagrangian mean curvature flow, Singularities · Same authors Jason D. Lotay, Goncalo Oliveira · Shared terms “neck pinch”, “pinch singularity”

  2. Generic neck pinch singularities along 2D Lagrangian mean curvature flow

    Gábor SzékelyhidiWider flows PDF

    Topics Lagrangian mean curvature flow, Singularities · Shared terms “neck pinch”, “pinch singularity”, “nondegenerate neck”

  3. Neck pinches along the Lagrangian mean curvature flow of surfaces

    Jason D. Lotay, Felix Schulze and Gábor SzékelyhidiWider flows PDF

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

  4. Ancient solutions and translators of Lagrangian mean curvature flow

    Jason D. Lotay, Felix Schulze and Gábor SzékelyhidiWider flows PDF

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

  5. Infinite-Time Singularities with Vanishing Mean Curvature for Lagrangian Mean Curvature Flow in Gibbons–Hawking Spaces

    Ping-Hung Lee and Chung-Jun TsaiWider flows PDF

    Topics Lagrangian mean curvature flow, Singularities · Shared terms “curvature blow”, “becomes unbounded”

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

    Sourav GhoshWider flows PDF

    Topics Lagrangian mean curvature flow, Singularities

  7. Singularity formations in Lagrangian mean curvature flow

    Yang Li and Gábor SzékelyhidiWider flows PDF

    Topics Lagrangian mean curvature flow, Singularities

  8. Nondegenerate neck pinches along the mean curvature flow

    Gábor SzékelyhidiWider flows PDF

    Topics Singularities · Shared terms “nondegenerate neck”, “neck pinch”, “pinch singularity”

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. 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 usually cannot run forever: curvature blows up in finite time, 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.