Paper · Oct 12, 2019

Ancient Solution of Mean Curvature Flow in Space Forms

Li Lei, Hongwei Xu, Entao Zhao

math.DGWider flowsarXiv:1910.05496v2
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Abstract

In this paper we investigate the rigidity of ancient solutions of the mean curvature flow with arbitrary codimension in space forms. We first prove that under certain sharp asymptotic pointwise curvature pinching condition the ancient solution in a sphere is either a shrinking spherical cap or a totally geodesic sphere. Then we show that under certain pointwise curvature pinching condition the ancient solution in a hyperbolic space is a family of shrinking spheres. We also obtain a rigidity result for ancient solutions in a nonnegatively curved space form under an asymptotic integral curvature pinching condition.

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Submitted
Oct 12, 2019
Last revised
Nov 28, 2019 (v2)
Subjects
math.DG

Comments: 23 pages

Mean curvature flowAncient solutionsCurvature pinchingSphere theoremsStability & 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 2016, 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 Mean curvature flow of arbitrary codimension in complex projective spaces (Li Lei and Hongwei Xu, 2016)1Lei 2016 Sharp one-sided curvature estimates for mean curvature flow and applications (Mat Langford, 2017)8Langford 2017 Pinched Ancient Solutions to the High Codimension Mean Curvature Flow (Stephen Lynch and Huy The Nguyen, 2017)4Lynch 2017 Ancient solutions of geometric flows with curvature pinching (Susanna Risa and Carlo Sinestrari, 2017)2Risa 2017 New Developments in Mean Curvature Flow of Arbitrary Codimension Inspired By Yau Rigidity Theory (Li Lei and Hong-Wei Xu, 2020)5Lei 2020 Surfaces pinched by normal curvature for mean curvature flow in space forms (Dong Pu, Jingjing Su and Hongwei Xu, 2020)3Pu 2020 Classification of quadratically pinched self-shrinkers in higher codimension (Debora Impera, Michele Rimoldi and Francesco Ruatta, 2026)7Impera 2026 Local Convexity and Planarity Estimates for Mean Curvature Flow (Liang Cheng, 2026)6Cheng 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Mean curvature flow of arbitrary codimension in complex projective spaces

    Li Lei and Hongwei XuWider flows

    Topics Curvature pinching, Sphere theorems · Same authors Li Lei, Hongwei Xu · Shared terms “pinching condition”, “totally geodesic”

  2. Ancient solutions of geometric flows with curvature pinching

    Susanna Risa and Carlo SinestrariWider flows

    Topics Curvature pinching, Ancient solutions, Stability & uniqueness · Shared terms “pinching condition”, “shrinking sphere”

  3. Surfaces pinched by normal curvature for mean curvature flow in space forms

    Dong Pu, Jingjing Su and Hongwei XuWider flows

    Topics Curvature pinching, Sphere theorems · Same author Hongwei Xu · Shared terms “space form”, “pinching condition”

  4. Pinched Ancient Solutions to the High Codimension Mean Curvature Flow

    Stephen Lynch and Huy The NguyenWider flows

    Topics Curvature pinching, Ancient solutions · Shared terms “pinching condition”, “shrinking sphere”

  5. New Developments in Mean Curvature Flow of Arbitrary Codimension Inspired By Yau Rigidity Theory

    Li Lei and Hong-Wei XuWider flows

    Topics Sphere theorems, Stability & uniqueness · Same author Li Lei

  6. Local Convexity and Planarity Estimates for Mean Curvature Flow

    Liang ChengWider flows

    Topics Curvature pinching, Ancient solutions · Shared terms “pinching condition”, “pointwise”

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

    Debora Impera, Michele Rimoldi and Francesco RuattaWider flows

    Topics Curvature pinching, Ancient solutions · Shared terms “pinching condition”

  8. Sharp one-sided curvature estimates for mean curvature flow and applications

    Mat LangfordWider flows

    Topics Curvature pinching, Ancient solutions · Shared terms “shrinking sphere”

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 in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) 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) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988)

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.