Paper · Dec 12, 2018

Ancient solutions for Andrews' hypersurface flow

Peng Lu, Jiuru Zhou

math.DGarXiv:1812.04926
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Abstract

We construct the ancient solutions of the hypersurface flows in Euclidean spaces studied by B. Andrews in 1994. As time the solutions collapse to a round point where is the singular time. But as the solutions become more and more oval. Near the center the appropriately-rescaled pointed Cheeger-Gromov limits are round cylinder solutions , . These results are the analog of the corresponding results in Ricci flow () and mean curvature flow.

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Submitted
Dec 12, 2018
Subjects
math.DG
Ricci flowMean curvature flowAncient solutions

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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 Ancient Ricci Flow Solutions on Bundles (Peng Lu and Y. K. Wang, 2016)4Lu 2016 Ancient Solutions on Bundles with Non-abelian Structural Group (Peng Lu and Y. K. Wang, 2016)2Lu 2016 New proofs of Perelman's theorem on shrinking Breathers in Ricci flow (Peng Lu and Yu Zheng, 2017)3Lu 2017 Ancient solutions to the Ricci flow in dimension 3 (S. Brendle, 2018)5Brendle 2018 Ancient solutions to the Kähler Ricci flow (Yu Li, 2020)6Li 2020 Uniqueness of compact ancient solutions to the higher dimensional Ricci flow (Simon Brendle, Panagiota Daskalopoulos, Keaton Naff et al., 2021)8Brendle 2021 Uniqueness of convex ancient solutions to hypersurface flows (Stephen Lynch, 2021)1Lynch 2021 κ-solutions with the round cylinder as an asymptotic shrinker (Aprameya Girish Hebbar, 2026)7Hebbar 2026
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  1. Uniqueness of convex ancient solutions to hypersurface flows

    Stephen LynchWider flows

    Topics Ancient solutions · Shared terms “hypersurface flows”

  2. Ancient Solutions on Bundles with Non-abelian Structural Group

    Peng Lu and Y. K. Wang

    Topics Ancient solutions · Same author Peng Lu

  3. New proofs of Perelman's theorem on shrinking Breathers in Ricci flow

    Peng Lu and Yu Zheng

    Topics Ancient solutions · Same author Peng Lu · Shared terms “rescaled”

  4. Ancient Ricci Flow Solutions on Bundles

    Peng Lu and Y. K. Wang

    Topics Ancient solutions · Same author Peng Lu

  5. Ancient solutions to the Ricci flow in dimension 3

    S. Brendle

    Topics Ancient solutions

  6. Ancient solutions to the Kähler Ricci flow

    Yu Li

    Topics Ancient solutions

  7. -solutions with the round cylinder as an asymptotic shrinker

    Aprameya Girish Hebbar

    Topics Ancient solutions · Shared terms “round cylinder”

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

    Simon Brendle, Panagiota Daskalopoulos, Keaton Naff et al.

    Topics Ancient solutions

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.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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)

In the guidebook

  1. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

  2. 12B.1
    κ-SolutionsBook 12B · κ-Solutions and Surgery

    Ancient, nonnegatively curved, noncollapsed: the models for singularities.

  3. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.