Paper · Oct 23, 2026

Instability of mean curvature flow in expander coordinates

Ben S. Lambert, Oliver C. Schnürer

math.DGarXiv:2610.32819
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Abstract

We study rotationally symmetric graphical solutions to mean curvature flow that are initially close to a homothetic reference expander, in particular near infinity. Based on a computer algebra aided proof, we show that under appropriate conditions, at large times, the solutions are closer to a different expander than to the reference expander.

Comments: 48 pages, 9 figures, includes sagemath code

Mean curvature flowRicci solitonsRotational symmetry & warped products

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 Existence of singular rotationally symmetric gradient Ricci solitons in higher dimensions (Kin Ming Hui, 2021)2Hui 2021 A note on gradient Ricci soliton warped metrics (José N. V. Gomes, Marcus A. M. Marrocos and Adrian V. C. Ribeiro, 2021)3Gomes 2021 Uniqueness of entire graphs evolving by Mean Curvature flow (Panagiota Daskalopoulos and Mariel Saez, 2021)5Daskalopoulo 2021 Conformal solitons for the mean curvature flow in hyperbolic space (Luciano Mari, Jose Danuso Rocha de Oliveira, Andreas Savas-Halilaj et al., 2023)7Mari 2023 Exponential Asymptotics of Ricci Flow Neckpinch (Hamidreza Mahmoudian, 2023)8Mahmoudian 2023 Isotropy invariant graphical mean curvature flows in warped products (Naotoshi Fujihara and Naoyuki Koike, 2024)6Fujihara 2024 Existence proofs for rotationally symmetric translating solutions to mean curvature flow (Hakar Raji and Oliver C. Schnürer, 2025)1Raji 2025 Self-expanders of positive genus (Guanhua Shao and Jiahua Zou, 2025)4Shao 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Existence proofs for rotationally symmetric translating solutions to mean curvature flow

    Hakar Raji and Oliver C. SchnürerWider flows PDF

    Topics Rotational symmetry & warped products · Same author Oliver C. Schnürer · Shared terms “rotationally symmetric”

  2. Existence of singular rotationally symmetric gradient Ricci solitons in higher dimensions

    Kin Ming Hui PDF

    Topics Rotational symmetry & warped products, Ricci solitons · Shared terms “rotationally symmetric”

  3. A note on gradient Ricci soliton warped metrics

    José N. V. Gomes, Marcus A. M. Marrocos and Adrian V. C. Ribeiro PDF

    Topics Rotational symmetry & warped products, Ricci solitons · Shared terms “appropriate”

  4. Self-expanders of positive genus

    Guanhua Shao and Jiahua ZouWider flows PDF

    Topics Rotational symmetry & warped products · Shared terms “expander”, “rotationally symmetric”

  5. Uniqueness of entire graphs evolving by Mean Curvature flow

    Panagiota Daskalopoulos and Mariel SaezWider flows PDF

    Topics Rotational symmetry & warped products · Shared terms “rotationally symmetric”, “graphical”

  6. Isotropy invariant graphical mean curvature flows in warped products

    Naotoshi Fujihara and Naoyuki KoikeWider flows PDF

    Topics Rotational symmetry & warped products · Shared terms “graphical”

  7. Conformal solitons for the mean curvature flow in hyperbolic space

    Luciano Mari, Jose Danuso Rocha de Oliveira, Andreas Savas-Halilaj et al. PDF

    Topics Rotational symmetry & warped products, Ricci solitons · Shared terms “expander”, “rotationally symmetric”, “appropriate”

  8. Exponential Asymptotics of Ricci Flow Neckpinch

    Hamidreza Mahmoudian PDF

    Topics Rotational symmetry & warped products · Shared terms “rotationally symmetric”

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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

In the guidebook

  1. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

  3. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.