Paper · Jun 28, 2022

A note on Alexandrov immersed mean curvature flow

Ben Lambert, Elena Mäder-Baumdicker

math.DGWider flowsarXiv:2206.14060
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Abstract

We demonstrate that the property of being Alexandrov immersed is preserved along mean curvature flow. Furthermore, we demonstrate that mean curvature flow techniques for mean convex embedded flows such as noncollapsing and gradient estimates also hold in this setting. We also indicate the necessary modifications to the work of Brendle–Huisken to allow for mean curvature flow with surgery for the Alexandrov immersed, -dimensional setting.

Comments: 19 pages

Mean curvature flowSurgery(Non)collapsing

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 Notes on Perelman's papers (Bruce Kleiner and John Lott, 2006)3Kleiner 2006 Quadratically pinched submanifolds of the sphere via mean curvature flow with surgery (Mat Langford, Stephen Lynch and Huy The Nguyen, 2021)7Langford 2021 Nonlocal estimates for the Volume Preserving Mean Curvature Flow and applications (Ben Lambert and Elena Mäder-Baumdicker, 2022)1Lambert 2022 Passing through nondegenerate singularities in mean curvature flows (Ao Sun, Zhihan Wang and Jinxin Xue, 2025)4Sun 2025 Towards a Taub-Bolt to Taub-NUT via Ricci flow with surgery (John Hughes, 2025)8Hughes 2025 Nondegenerate neck pinches along the mean curvature flow (Gábor Székelyhidi, 2026)5Székelyhidi 2026 Mean convex flows with surgery (Gábor Székelyhidi, 2026)2Székelyhidi 2026 Positive τ-bi-Ricci curvature and Mean curvature flow with surgery in hyperbolic space (Tianci Luo, Yong Wei and Rong Zhou, 2026)6Luo 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Nonlocal estimates for the Volume Preserving Mean Curvature Flow and applications

    Ben Lambert and Elena Mäder-BaumdickerWider flows PDF

    Topics Mean curvature flow · Same authors Ben Lambert, Elena Mäder-Baumdicker · Shared terms “demonstrate”

  2. Mean convex flows with surgery

    Gábor SzékelyhidiWider flows PDF

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

  3. Notes on Perelman's papers

    Bruce Kleiner and John LottThe canon PDF

    Topics Surgery

  4. Passing through nondegenerate singularities in mean curvature flows

    Ao Sun, Zhihan Wang and Jinxin XueWider flows PDF

    Topics Surgery

  5. Nondegenerate neck pinches along the mean curvature flow

    Gábor SzékelyhidiWider flows PDF

    Topics Surgery

  6. Positive -bi-Ricci curvature and Mean curvature flow with surgery in hyperbolic space

    Tianci Luo, Yong Wei and Rong ZhouWider flows PDF

    Topics Surgery · Shared terms “immersed”, “preserved”

  7. Quadratically pinched submanifolds of the sphere via mean curvature flow with surgery

    Mat Langford, Stephen Lynch and Huy The NguyenWider flows PDF

    Topics Surgery · Shared terms “immersed”

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

    John Hughes PDF

    Topics Surgery

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) 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) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982)

In the guidebook

  1. 12B.11
    Noncollapsing After SurgeryBook 12B · κ-Solutions and Surgery

    Perelman II Lemma 5.2 proved: admissible curves, the evolution of surgery caps, the length estimate for curves that come near a surgery, and the reduced-volume argument in a space-time with surgeries (Morgan–Tian chapter 16).

  2. 11B.7
    Hamilton’s Program in 2002Book 11B · Solitons, Compactness and Singularities

    What was proved, what was missing, and where Perelman came in.

  3. 12B.5
    The Canonical Neighbourhood TheoremBook 12B · κ-Solutions and Surgery

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