Paper · Mar 11, 2026

Nondegenerate neck pinches along the mean curvature flow

Gábor Székelyhidi

math.DGWider flowsarXiv:2603.11022v2
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Abstract

We show that for generic smooth compact initial surfaces the mean curvature flow in has spherical or nondegenerate neck pinch singularities at the first singular time. In particular the singularities at the first singular time are isolated in spacetime. As an application we give a new approach to constructing a mean curvature flow with surgery for smooth compact initial surfaces in .

Comments: v2: 30 pages, made some corrections, added application to mean curvature flow with surgery

Mean curvature flowSingularitiesSurgery

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 Neck pinches along the Lagrangian mean curvature flow of surfaces (Jason D. Lotay, Felix Schulze and Gábor Székelyhidi, 2022)1Lotay 2022 Neck pinch singularities and Joyce conjectures in Lagrangian mean curvature flow with circle symmetry (Jason D. Lotay and Goncalo Oliveira, 2023)7Lotay 2023 Singularity formations in Lagrangian mean curvature flow (Yang Li and Gábor Székelyhidi, 2024)8Li 2024 Passing through nondegenerate singularities in mean curvature flows (Ao Sun, Zhihan Wang and Jinxin Xue, 2025)5Sun 2025 Generic neck pinch singularities along 2D Lagrangian mean curvature flow (Gábor Székelyhidi, 2026)2Székelyhidi 2026 Mean convex flows with surgery (Gábor Székelyhidi, 2026)3Székelyhidi 2026 The mean curvature blows up at nondegenerate neck pinch singularities of Lagrangian mean curvature flow (Jason D. Lotay and Goncalo Oliveira, 2026)4Lotay 2026 Generic mean curvature flow has finitely many singularities (J. M. Daniels-Holgate, Or Hershkovits and Gábor Székelyhidi, 2026)6Daniels-Holg 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Neck pinches along the Lagrangian mean curvature flow of surfaces

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

    Topics Singularities · Same author Gábor Székelyhidi · Shared terms “pinche along”, “neck pinches”, “singular time”

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

    Gábor SzékelyhidiWider flows PDF

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

  3. Mean convex flows with surgery

    Gábor SzékelyhidiWider flows PDF

    Topics Surgery, Singularities · Same author Gábor Székelyhidi · Shared terms “nondegenerate”

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

    Jason D. Lotay and Goncalo OliveiraWider flows PDF

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

  5. Passing through nondegenerate singularities in mean curvature flows

    Ao Sun, Zhihan Wang and Jinxin XueWider flows PDF

    Topics Surgery, Singularities · Shared terms “nondegenerate”, “isolated”

  6. Generic mean curvature flow has finitely many singularities

    J. M. Daniels-Holgate, Or Hershkovits and Gábor SzékelyhidiWider flows PDF

    Topics Singularities · Same author Gábor Székelyhidi · Shared terms “compact initial”, “initial surface”

  7. Neck pinch singularities and Joyce conjectures in Lagrangian mean curvature flow with circle symmetry

    Jason D. Lotay and Goncalo OliveiraWider flows PDF

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

  8. Singularity formations in Lagrangian mean curvature flow

    Yang Li and Gábor SzékelyhidiWider flows PDF

    Topics Singularities · Same author Gábor Székelyhidi

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) 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) Surgery Stop the flow just before a neck pinches, cut along the thin neck, glue standard caps onto the two ends, and restart. Start with: Four-manifolds with positive isotropic curvature (Hamilton, 1997)

In the guidebook

  1. 12B.5
    The Canonical Neighbourhood TheoremBook 12B · κ-Solutions and Surgery

    Every high-curvature point looks like a neck or a cap.

  2. 12B.7
    The Singular Time: Horns and Strong NecksBook 12B · κ-Solutions and Surgery

    The limit metric at a singular time, the set Ω where curvature stays bounded, ε-horns and their ends, and strong δ-necks deep in a horn, following Perelman II §3 and Morgan–Tian chapter 11.

  3. 12B.12
    Canonical Neighbourhoods After SurgeryBook 12B · κ-Solutions and Surgery

    The strong canonical neighbourhood assumption justified across surgeries: the blow-up argument, the three cases for a limit that meets a surgery, and the choice of δ (Perelman II §5; Morgan–Tian chapter 17).