Paper · Dec 21, 2022

On the Existence and Uniqueness of Ancient Rescaled Mean Curvature Flows

Letian Chen

math.DGWider flowsarXiv:2212.10798
View PDF View HTML arXiv page

As on arXiv

Abstract

We show existence of ancient solutions to the rescaled mean curvature flow starting from a given asymptotically conical self-expander. These are examples of mean curvature flows coming out of cones that are not self-similar. We also show a strong uniqueness theorem when the cone is generic and use it to classify mean curvature flows coming out of generic cones of small entropy in low dimensions.

View PDF View HTML arXiv page
Submitted
Dec 21, 2022
Subjects
math.DG, math.AP

Comments: 31 pages

Mean curvature flowAncient solutionsSelf-shrinkers & translatorsWeak & singular flowsStability & uniquenessBoundaries & non-compact

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 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 Rotational symmetry of solutions of mean curvature flow coming out of a double cone II (Letian Chen, 2022)1Chen 2022 Dynamics of Convex Mean Curvature Flow (Sigurd Angenent, Panagiota Daskalopoulos and Natasa Sesum, 2023)3Angenent 2023 Closed mean curvature flows with asymptotically conical singularities (Tang-Kai Lee and Xinrui Zhao, 2024)8Lee 2024 A rigidity theorem of ancient solutions to the mean curvature flow in codimension one (Qun Chen and Hongbing Qiu, 2024)4Chen 2024 Backwards uniqueness for Mean curvature flow with asymptotically conical singularities (J. M. Daniels-Holgate and Or Hershkovits, 2025)5Daniels-Holg 2025 Self-expanders of positive genus (Guanhua Shao and Jiahua Zou, 2025)7Shao 2025 Convexity of mean convex asymptotically conical self-expanders to the mean curvature flow (Junming Xie, 2025)2Xie 2025 Mean curvature flow near a peanut solution (Sigurd Angenent, Panagiota Daskalopoulos and Natasa Sesum, 2025)6Angenent 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Rotational symmetry of solutions of mean curvature flow coming out of a double cone II

    Letian ChenWider flows

    Topics Weak & singular flows · Same author Letian Chen · Shared terms “flow coming”

  2. Convexity of mean convex asymptotically conical self-expanders to the mean curvature flow

    Junming XieWider flows

    Topics Self-shrinkers & translators, Boundaries & non-compact · Shared terms “conical self”, “self expanders”, “asymptotically conical”

  3. Dynamics of Convex Mean Curvature Flow

    Sigurd Angenent, Panagiota Daskalopoulos and Natasa SesumWider flows

    Topics Ancient solutions, Weak & singular flows, Boundaries & non-compact · Shared terms “rescaled mean”

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

    Qun Chen and Hongbing QiuWider flows

    Topics Ancient solutions, Boundaries & non-compact, Stability & uniqueness

  5. Backwards uniqueness for Mean curvature flow with asymptotically conical singularities

    J. M. Daniels-Holgate and Or HershkovitsWider flows

    Topics Self-shrinkers & translators, Boundaries & non-compact, Stability & uniqueness · Shared terms “asymptotically conical”

  6. Mean curvature flow near a peanut solution

    Sigurd Angenent, Panagiota Daskalopoulos and Natasa SesumWider flows

    Topics Ancient solutions, Self-shrinkers & translators, Stability & uniqueness · Shared terms “rescaled”, “flow starting”

  7. Self-expanders of positive genus

    Guanhua Shao and Jiahua ZouWider flows

    Topics Self-shrinkers & translators, Weak & singular flows · Shared terms “self expanders”

  8. Closed mean curvature flows with asymptotically conical singularities

    Tang-Kai Lee and Xinrui Zhao

    Topics Self-shrinkers & translators, Weak & singular flows, Boundaries & non-compact · Shared terms “conical self”, “asymptotically conical”, “flow starting”

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) 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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

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

    Comparing curvature across space and time.

  3. 12B.1
    κ-SolutionsBook 12B · κ-Solutions and Surgery

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