Paper · May 31, 2026

Mean convex flows with surgery

Gábor Székelyhidi

math.DGWider flowsarXiv:2606.00964
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Abstract

We construct a mean curvature flow with surgery starting with any compact mean convex hypersurface in , extending previous results of Huisken-Sinestrari, Brendle-Huisken, and Haslhofer-Kleiner for 2-convex flows. In contrast with previous constructions of mean curvature flow with surgery, the topological surgeries are performed by the flow itself through nondegenerate cylindrical singularities. At the same time the flow needs to be slightly adjusted at finitely many smooth times.

Comments: 56 pages

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 Canonical surgeries in rotationally invariant Ricci flow (Timothy Buttsworth, Maximilien Hallgren and Yongjia Zhang, 2022)6Buttsworth 2022 A note on Alexandrov immersed mean curvature flow (Ben Lambert and Elena Mäder-Baumdicker, 2022)4Lambert 2022 Cobordism, Singularities and the Ricci Flow Conjecture (David Martín Velázquez, Davide De Biasio and Dieter Lust, 2022)8Velázquez 2022 Free boundary flow with surgery (Robert Haslhofer, 2023)7Haslhofer 2023 Passing through nondegenerate singularities in mean curvature flows (Ao Sun, Zhihan Wang and Jinxin Xue, 2025)2Sun 2025 Towards a Taub-Bolt to Taub-NUT via Ricci flow with surgery (John Hughes, 2025)5Hughes 2025 Nondegenerate neck pinches along the mean curvature flow (Gábor Székelyhidi, 2026)1Székelyhidi 2026 Generic mean curvature flow has finitely many singularities (J. M. Daniels-Holgate, Or Hershkovits and Gábor Székelyhidi, 2026)3Daniels-Holg 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Nondegenerate neck pinches along the mean curvature flow

    Gábor SzékelyhidiWider flows PDF

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

  2. Passing through nondegenerate singularities in mean curvature flows

    Ao Sun, Zhihan Wang and Jinxin XueWider flows PDF

    Topics Surgery, Singularities · Shared terms “nondegenerate cylindrical”, “cylindrical singularities”

  3. 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 “finitely”, “mean convex”, “cylindrical singularities”

  4. A note on Alexandrov immersed mean curvature flow

    Ben Lambert and Elena Mäder-BaumdickerWider flows PDF

    Topics Surgery · Shared terms “mean convex”, “Huisken”, “Brendle”

  5. Towards a Taub-Bolt to Taub-NUT via Ricci flow with surgery

    John Hughes PDF

    Topics Surgery, Singularities

  6. Canonical surgeries in rotationally invariant Ricci flow

    Timothy Buttsworth, Maximilien Hallgren and Yongjia Zhang PDF

    Topics Surgery, Singularities · Shared terms “surgery starting”, “Kleiner”, “Haslhofer”

  7. Free boundary flow with surgery

    Robert HaslhoferWider flows PDF

    Topics Surgery, Singularities · Shared terms “mean convex”

  8. Cobordism, Singularities and the Ricci Flow Conjecture

    David Martín Velázquez, Davide De Biasio and Dieter Lust PDF

    Topics Surgery, 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) 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).