Paper · Feb 17, 2026

Generic neck pinch singularities along 2D Lagrangian mean curvature flow

Gábor Székelyhidi

math.DGWider flowsarXiv:2602.15771
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Abstract

We introduce a notion of nondegenerate neck pinch singularity along the Lagrangian mean curvature flow of surfaces in a Calabi-Yau surface. We show that such singularities can occur, are stable under small perturbations, and any neck pinch singularity can be perturbed to such a nondegenerate singularity near the singular time. Using this we answer some questions raised by Neves and Joyce. We also introduce nondegenerate teardrop singularities and show that these cannot occur for embedded flows.

Comments: 26 pages

Mean curvature flowLagrangian mean curvature flowRicci-flat & Calabi–YauSingularitiesKähler geometry

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)6Lotay 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)2Lotay 2023 Singularity formations in Lagrangian mean curvature flow (Yang Li and Gábor Székelyhidi, 2024)4Li 2024 Lagrangian mean curvature flow of surfaces with mean curvature bound (Sourav Ghosh, 2025)8Ghosh 2025 Nondegenerate neck pinches along the mean curvature flow (Gábor Székelyhidi, 2026)5Székelyhidi 2026 The mean curvature blows up at nondegenerate neck pinch singularities of Lagrangian mean curvature flow (Jason D. Lotay and Goncalo Oliveira, 2026)1Lotay 2026 Lagrangian mean curvature flow in the Kummer K3 surface (Carlos Alberto Ochoa Flores, 2026)7Flores 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The mean curvature blows up at nondegenerate neck pinch singularities of Lagrangian mean curvature flow

    Jason D. Lotay and Goncalo OliveiraWider flows PDF

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

  2. 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, Ricci-flat & Calabi–Yau, Kähler geometry · Shared terms “neck pinch”, “pinch singularity”, “Joyce”

  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, Ricci-flat & Calabi–Yau, Kähler geometry · Same author Gábor Székelyhidi · Shared terms “Yau surface”, “singular time”

  4. Singularity formations in Lagrangian mean curvature flow

    Yang Li and Gábor SzékelyhidiWider flows PDF

    Topics Lagrangian mean curvature flow, Singularities · Same author Gábor Székelyhidi · Shared terms “singularities along”, “Joyce”

  5. Nondegenerate neck pinches along the mean curvature flow

    Gábor SzékelyhidiWider flows PDF

    Topics Singularities · Same author Gábor Székelyhidi · Shared terms “nondegenerate neck”, “neck pinch”, “pinch singularity”

  6. 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 Gábor Székelyhidi

  7. Lagrangian mean curvature flow in the Kummer K3 surface

    Carlos Alberto Ochoa FloresWider flows PDF

    Topics Lagrangian mean curvature flow, Ricci-flat & Calabi–Yau · Shared terms “perturbation”

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

    Sourav GhoshWider flows PDF

    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. 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) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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)

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