Paper · Oct 1, 2017

Ancient solutions of geometric flows with curvature pinching

Susanna Risa, Carlo Sinestrari

math.DGWider flowsarXiv:1710.00361v3
View PDF View HTML arXiv page

As on arXiv

Abstract

We prove rigidity theorems for ancient solutions of geometric flows of immersed submanifolds. Specifically, we find pinching conditions on the second fundamental form that characterize the shrinking sphere among compact ancient solutions for the mean curvature flow in codimension greater than one, and for some nonlinear curvature flows of hypersurfaces.

View PDF View HTML arXiv page
Submitted
Oct 1, 2017
Last revised
May 22, 2018 (v3)
Subjects
math.DG, math.AP
DOI
10.1007/s12220-018-0036-0

Comments: Minor changes, bibliography updated. To appear on The Journal of Geometric Analysis. The final version is available online at https://doi.org/10.1007/s12220-018-0036-0

Mean curvature flowOther curvature flowsAncient solutionsCurvature 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 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 Sharp one-sided curvature estimates for fully nonlinear curvature flows and applications to ancient solutions (Mat Langford and Stephen Lynch, 2017)5Langford 2017 Pinched Ancient Solutions to the High Codimension Mean Curvature Flow (Stephen Lynch and Huy The Nguyen, 2017)4Lynch 2017 Strong spherical rigidity of ancient solutions of expansive curvature flows (Susanna Risa and Carlo Sinestrari, 2019)1Risa 2019 Ancient Solution of Mean Curvature Flow in Space Forms (Li Lei, Hongwei Xu and Entao Zhao, 2019)2Lei 2019 Sharp pinching estimates for mean curvature flow in the sphere (Mat Langford and Huy The Nguyen, 2020)7Langford 2020 Non-homothetic convex ancient solutions for flows by high powers of curvature (Susanna Risa and Carlo Sinestrari, 2022)3Risa 2022 A rigidity theorem of ancient solutions to the mean curvature flow in codimension one (Qun Chen and Hongbing Qiu, 2024)6Chen 2024 Classification of quadratically pinched self-shrinkers 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. Strong spherical rigidity of ancient solutions of expansive curvature flows

    Susanna Risa and Carlo SinestrariWider flows

    Topics Ancient solutions, Other curvature flows, Stability & uniqueness · Same authors Susanna Risa, Carlo Sinestrari · Shared terms “geometric flows”

  2. Ancient Solution of Mean Curvature Flow in Space Forms

    Li Lei, Hongwei Xu and Entao ZhaoWider flows

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

  3. Non-homothetic convex ancient solutions for flows by high powers of curvature

    Susanna Risa and Carlo SinestrariWider flows

    Topics Ancient solutions, Other curvature flows · Same authors Susanna Risa, Carlo Sinestrari

  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”, “compact ancient”, “shrinking sphere”

  5. Sharp one-sided curvature estimates for fully nonlinear curvature flows and applications to ancient solutions

    Mat Langford and Stephen LynchWider flows

    Topics Curvature pinching, Ancient solutions, Other curvature flows · Shared terms “nonlinear curvature”, “shrinking sphere”

  6. A rigidity theorem of ancient solutions to the mean curvature flow in codimension one

    Qun Chen and Hongbing QiuWider flows

    Topics Ancient solutions, Stability & uniqueness · Shared terms “rigidity theorem”, “fundamental form”

  7. Sharp pinching estimates for mean curvature flow in the sphere

    Mat Langford and Huy The NguyenWider flows

    Topics Curvature pinching, Ancient solutions, Stability & uniqueness · Shared terms “rigidity theorem”, “fundamental form”

  8. 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”, “fundamental form”, “Specifically”

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) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) 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)

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.