Paper · Feb 13, 2025

On mean curvature flow solitons in the sphere

Marco Magliaro, Luciano Mari, Fernanda Roing, Andreas Savas-Halilaj

math.DGWider flowsarXiv:2502.09199v2
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Abstract

In this paper, we consider soliton solutions of the mean curvature flow in the unit sphere moving along the integral curves of the Hopf unit vector field. While such solitons must necessarily be minimal if compact, we produce a non-minimal, complete example with topology . The example wraps around a Clifford torus along each end, it has reflection and rotational symmetry and its mean curvature changes sign on each end. Indeed, we prove that a complete 2-dimensional soliton with non-negative mean curvature outside a compact set must be a covering of a Clifford torus. Concluding, we obtain a pinching theorem under suitable conditions on the second fundamental form.

Comments: To appear in Revista Matemática Iberoamericana

Mean curvature flowCurvature pinchingSurfacesRotational symmetry & warped products

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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 Singularity Models for High Codimension Mean Curvature Flow in Riemannian Manifolds (Artemis A. Vogiatzi and Huy T. Nguyen, 2023)6Vogiatzi 2023 Conformal solitons for the mean curvature flow in hyperbolic space (Luciano Mari, Jose Danuso Rocha de Oliveira, Andreas Savas-Halilaj et al., 2023)1Mari 2023 A gap Theorem on closed self-shrinkers of mean curvature flow (Yuhang Zhao, 2025)3Zhao 2025 A mean curvature flow approach to Hamilton's pinching theorem (Liang Cheng and Zhenyu Lu, 2026)2Cheng 2026 Singular Rotational Self-Similar Tori for Odd σₖ-Curvature Flows (Haoxuan Cheng, Junqi Lai and Guoxin Wei, 2026)7Cheng 2026 The rigidity theorems for complete self-expander of mean curvature flow (Zhi Li and Guoxin Wei, 2026)5Li 2026 The planarity estimate in dimensions five and six (Stephen Lynch, 2026)8Lynch 2026 Tensorial pinching by deforming curvature cones in higher codimension mean curvature flow (Tsz-Kiu Aaron Chow, 2026)4Chow 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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 · Same authors Luciano Mari, Andreas Savas-Halilaj

  2. A mean curvature flow approach to Hamilton's pinching theorem

    Liang Cheng and Zhenyu LuWider flows PDF

    Topics Curvature pinching · Shared terms “pinching theorem”

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

    Yuhang ZhaoWider flows PDF

    Topics Curvature pinching · Shared terms “pinching theorem”, “fundamental form”

  4. Tensorial pinching by deforming curvature cones in higher codimension mean curvature flow

    Tsz-Kiu Aaron Chow PDF

    Topics Curvature pinching · Shared terms “fundamental form”

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

    Zhi Li and Guoxin WeiWider flows PDF

    Topics Curvature pinching · Shared terms “fundamental form”

  6. Singularity Models for High Codimension Mean Curvature Flow in Riemannian Manifolds

    Artemis A. Vogiatzi and Huy T. NguyenWider flows PDF

    Topics Curvature pinching

  7. Singular Rotational Self-Similar Tori for Odd -Curvature Flows

    Haoxuan Cheng, Junqi Lai and Guoxin WeiWider flows PDF

    Topics Rotational symmetry & warped products

  8. The planarity estimate in dimensions five and six

    Stephen LynchWider flows PDF

    Topics Curvature pinching · Shared terms “fundamental form”

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) 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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… 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. 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.