Paper · Mar 30, 2018

Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions

S. Brendle, K. Choi

math.DGWider flowsarXiv:1804.00018v3
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Abstract

In this paper, we consider noncompact ancient solutions to the mean curvature flow in () which are strictly convex, uniformly two-convex, and noncollapsed. We prove that such an ancient solution is a rotationally symmetric translating soliton.

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Submitted
Mar 30, 2018
Last revised
Jun 28, 2022 (v3)
Subjects
math.DG
Journal
Geom. Topol. 25 (2021) 2195-2234
DOI
10.1007/s00222-019-00859-4

Comments: This is the final published version

Mean curvature flowAncient solutionsSelf-shrinkers & translators(Non)collapsingStability & uniquenessBoundaries & non-compactRotational symmetry & warped products

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 Uniqueness of convex ancient solutions to mean curvature flow in ℝ³ (S. Brendle and K. Choi, 2017)1Brendle 2017 Ancient solutions to the Ricci flow in higher dimensions (Xiaolong Li and Yongjia Zhang, 2018)6Li 2018 Rotational symmetry of ancient solutions to the Ricci flow in dimension 3 – The compact case (S. Brendle, 2019)4Brendle 2019 Rotational symmetry of ancient solutions to the Ricci flow in higher dimensions (Simon Brendle and Keaton Naff, 2020)2Brendle 2020 Uniqueness of compact ancient solutions to the higher dimensional Ricci flow (Simon Brendle, Panagiota Daskalopoulos, Keaton Naff et al., 2021)3Brendle 2021 Uniqueness of convex ancient solutions to hypersurface flows (Stephen Lynch, 2021)5Lynch 2021 Rotational symmetry of ancient solutions to fully nonlinear curvature flows (A. Cogo, S. Lynch and O. Vičánek Martínez, 2023)8Cogo 2023 Classification of ancient noncollapsed flows in ℝ⁴ (Kyeongsu Choi and Robert Haslhofer, 2024)7Choi 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Uniqueness of convex ancient solutions to mean curvature flow in

    S. Brendle and K. Choi

    Topics Rotational symmetry & warped products, (Non)collapsing, Ancient solutions · Same authors S. Brendle, K. Choi · Shared terms “convex ancient”, “noncompact ancient”, “noncollapsed”

  2. Rotational symmetry of ancient solutions to the Ricci flow in higher dimensions

    Simon Brendle and Keaton Naff

    Topics Rotational symmetry & warped products, (Non)collapsing, Ancient solutions · Same author S. Brendle · Shared terms “higher dimensions”, “noncompact ancient”, “noncollapsed”

  3. Uniqueness of compact ancient solutions to the higher dimensional Ricci flow

    Simon Brendle, Panagiota Daskalopoulos, Keaton Naff et al.

    Topics (Non)collapsing, Ancient solutions, Stability & uniqueness · Same author S. Brendle · Shared terms “higher dimensions”, “noncollapsed”

  4. Rotational symmetry of ancient solutions to the Ricci flow in dimension – The compact case

    S. Brendle

    Topics Rotational symmetry & warped products, Ancient solutions, Boundaries & non-compact · Same author S. Brendle · Shared terms “noncompact ancient”, “rotationally symmetric”

  5. Uniqueness of convex ancient solutions to hypersurface flows

    Stephen LynchWider flows

    Topics (Non)collapsing, Ancient solutions, Stability & uniqueness · Shared terms “convex ancient”

  6. Ancient solutions to the Ricci flow in higher dimensions

    Xiaolong Li and Yongjia Zhang

    Topics Rotational symmetry & warped products, (Non)collapsing, Ancient solutions · Shared terms “higher dimensions”, “noncollapsed”, “rotationally symmetric”

  7. Classification of ancient noncollapsed flows in

    Kyeongsu Choi and Robert HaslhoferWider flows

    Topics Rotational symmetry & warped products, (Non)collapsing, Ancient solutions · Same author K. Choi · Shared terms “noncollapsed”, “strictly convex”, “rotationally symmetric”

  8. Rotational symmetry of ancient solutions to fully nonlinear curvature flows

    A. Cogo, S. Lynch and O. Vičánek MartínezWider flows

    Topics Rotational symmetry & warped products, (Non)collapsing, Ancient solutions · Shared terms “symmetric translating”, “translating solitons”, “rotationally symmetric”

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) 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) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982)

In the guidebook

  1. 12B.4
    Compactness and Universal Estimates for κ-SolutionsBook 12B · κ-Solutions and Surgery

    Asymptotic volume zero, curvature controlled by volume, Perelman's Theorem 11.7, a universal κ, neck-like behaviour at infinity, and the constants that canonical neighbourhoods inherit.

  2. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  3. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.