Paper · Oct 19, 2026

The global structure of bi-rotational self-expanders

Gregŕio Silva Neto

math.DGWider flowsarXiv:2610.25833
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Abstract

We describe the self-expanders of the mean curvature flow in invariant under . Apart from the Clifford minimal cone, every such hypersurface is complete, proper and noncompact, and belongs to one of three families: two one-parameter families of embedded hypersurfaces diffeomorphic to or to , whose profile curves leave an axis, and a two-parameter family diffeomorphic to , parametrized by the point of the profile curve closest to the origin, which is the generic family: its set of initial conditions is open and of full measure in the phase space. Every end is asymptotically conical, with explicit decay rates, and every such hypersurface has Euclidean volume growth. A profile curve of the third family has at most one loop, and the point of the curve closest to the origin lies on that loop.

Comments: 27 pages, 3 figures

Mean curvature flowSelf-shrinkers & translatorsBoundaries & non-compactMinimal & CMC surfaces

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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 No breather theorems for the mean curvature flow (Liang Cheng and Yongjia Zhang, 2021)4Cheng 2021 Noncompact self-shrinkers for mean curvature flow with arbitrary genus (Reto Buzano, Huy The Nguyen and Mario B. Schulz, 2021)3Buzano 2021 Self-expanders to the mean curvature flow based on the generalized Lawson-Osserman cone (Chen-Kuan Lee, 2022)6Lee 2022 Poincaré inequality and topological rigidity of translators and self-expanders for the mean curvature flow (Debora Impera and Michele Rimoldi, 2023)7Impera 2023 Codimension two mean curvature flow of entire graphs (Andreas Savas-Halilaj and Knut Smoczyk, 2024)8Savas-Halila 2024 Convexity of mean convex asymptotically conical self-expanders to the mean curvature flow (Junming Xie, 2025)2Xie 2025 Invariant λ-translators in the Heisenberg group (Antonio Bueno, 2026)5Bueno 2026 The global structure of bi-rotational self-shrinkers (Gregório Silva Neto, 2026)1Neto 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The global structure of bi-rotational self-shrinkers

    Gregório Silva NetoWider flows PDF

    Topics Minimal & CMC surfaces, Self-shrinkers & translators, Boundaries & non-compact · Shared terms “global structure”, “rotational self”, “profile curve”

  2. Convexity of mean convex asymptotically conical self-expanders to the mean curvature flow

    Junming XieWider flows PDF

    Topics Self-shrinkers & translators, Boundaries & non-compact · Shared terms “self expanders”, “asymptotically conical”

  3. Noncompact self-shrinkers for mean curvature flow with arbitrary genus

    Reto Buzano, Huy The Nguyen and Mario B. SchulzWider flows PDF

    Topics Minimal & CMC surfaces, Self-shrinkers & translators, Boundaries & non-compact · Shared terms “asymptotically conical”

  4. No breather theorems for the mean curvature flow

    Liang Cheng and Yongjia ZhangWider flows PDF

    Topics Self-shrinkers & translators, Boundaries & non-compact · Shared terms “self expanders”, “Euclidean”

  5. Invariant -translators in the Heisenberg group

    Antonio BuenoWider flows PDF

    Topics Minimal & CMC surfaces, Self-shrinkers & translators

  6. Self-expanders to the mean curvature flow based on the generalized Lawson-Osserman cone

    Chen-Kuan LeeWider flows PDF

    Topics Self-shrinkers & translators · Shared terms “self expanders”

  7. Poincaré inequality and topological rigidity of translators and self-expanders for the mean curvature flow

    Debora Impera and Michele RimoldiWider flows PDF

    Topics Self-shrinkers & translators · Shared terms “self expanders”, “Euclidean”

  8. Codimension two mean curvature flow of entire graphs

    Andreas Savas-Halilaj and Knut SmoczykWider flows PDF

    Topics Self-shrinkers & translators, Boundaries & non-compact · Shared terms “self expanders”, “asymptotically conical”

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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) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989) Minimal & CMC surfaces Minimal surfaces are critical points of area, with zero mean curvature; constant mean curvature (CMC) surfaces are critical for area at fixed volume. Start with: On the structure of manifolds with positive scalar curvature (Schoen et al., 1979)

In the guidebook

  1. 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.