Paper · Sep 17, 2026

The rigidity theorems for complete self-expander of mean curvature flow

Zhi Li, Guoxin Wei

math.DGWider flowsarXiv:2609.20429v2
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Abstract

Our first main result is a rigidity theorem for complete self-expanders in the Euclidean space with higher codimension, assuming an integral cur More precisely, any smooth complete self-expander () that satisfies both and for a positive constant depending only o isometric to . Moreover, we show that the rigidity result persists when the pinching condition is expressed in terms of the trace-free second fundamental form.

Mean curvature flowSelf-shrinkers & translatorsCurvature pinchingStability & uniqueness

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 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)5Impera 2023 Gap theorems for complete self-shrinkers of r-mean curvature flows (Hilário Alencar, G. Pacelli Bessa and Gregório Silva Neto, 2024)1Alencar 2024 A gap Theorem on closed self-shrinkers of mean curvature flow (Yuhang Zhao, 2025)2Zhao 2025 Convexity of mean convex asymptotically conical self-expanders to the mean curvature flow (Junming Xie, 2025)7Xie 2025 Classification of quadratically pinched self-shrinkers in higher codimension (Debora Impera, Michele Rimoldi and Francesco Ruatta, 2026)4Impera 2026 The Rigidity Theorems for Self-Shrinkers in the Mean Curvature Flow (Fagui Li and Yuhang Zhao, 2026)3Li 2026 Classification of quadratically pinched translators in higher codimension (Debora Impera, Michele Rimoldi and Francesco Ruatta, 2026)8Impera 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Gap theorems for complete self-shrinkers of -mean curvature flows

    Hilário Alencar, G. Pacelli Bessa and Gregório Silva NetoWider flows PDF

    Topics Self-shrinkers & translators · Shared terms “complete self”, “fundamental form”

  2. A gap Theorem on closed self-shrinkers of mean curvature flow

    Yuhang ZhaoWider flows PDF

    Topics Curvature pinching, Self-shrinkers & translators · Shared terms “fundamental form”

  3. The Rigidity Theorems for Self-Shrinkers in the Mean Curvature Flow

    Fagui Li and Yuhang ZhaoWider flows PDF

    Topics Curvature pinching, Self-shrinkers & translators, Stability & uniqueness · Shared terms “rigidity theorem”

  4. Classification of quadratically pinched self-shrinkers in higher codimension

    Debora Impera, Michele Rimoldi and Francesco RuattaWider flows PDF

    Topics Curvature pinching, Self-shrinkers & translators · Shared terms “higher codimension”, “pinching condition”, “fundamental form”

  5. 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, Stability & uniqueness · Shared terms “self expanders”, “Euclidean space”

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

    Junming XieWider flows PDF

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

  8. Classification of quadratically pinched translators in higher codimension

    Debora Impera, Michele Rimoldi and Francesco RuattaWider flows PDF

    Topics Curvature pinching, Self-shrinkers & translators, Stability & uniqueness · Shared terms “higher codimension”, “rigidity theorem”, “pinching condition”

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) 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) Curvature pinching Curvature conditions that Ricci flow preserves and improves. Hamilton–Ivey pinching makes positive curvature dominate in 3D blow-ups, and invariant… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

In the guidebook

  1. 6A.9
    Reaction–Diffusion: Fronts, Patterns and PinchingBook 6A · The Heat Equation and Its Relatives

    Critical patch size, travelling fronts and their speeds, bistable waves and the Allen–Cahn equation, Turing patterns, and the curvature of a Ricci flow as a reaction–diffusion system.

  2. 11A.7
    Maximum Principles under Ricci FlowBook 11A · The Ricci Flow: Existence and Maximum Principles

    Scalar and tensor maximum principles, and preserved curvature conditions.

  3. 11A.10
    Hamilton–Ivey PinchingBook 11A · The Ricci Flow: Existence and Maximum Principles

    Why high curvature in dimension 3 is almost positive.