Paper · Nov 21, 2023

Curvature estimates of ancient solutions to the mean curvature flow of higher codimension with convex Gauss image

Hongbing Qiu, Y. L. Xin

math.DGWider flowsarXiv:2311.12400
PDF Abstract page

As on arXiv

Abstract

By carrying out refined curvature estimates, we prove better rigidity theorems of complete noncompact ancient solutions to the mean curvature flow in higher codimension under various Gauss image restriction.

Mean curvature flowAncient solutionsStability & 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 Ancient solutions to the Ricci flow in dimension 3 (S. Brendle, 2018)4Brendle 2018 A flow approach to the Musielak-Orlicz-Gauss image problem (Qi-Rui Li, Weimin Sheng, Deping Ye et al., 2021)6Li 2021 Quantitative long range curvature estimate for mean curvature flow (Jingze Zhu, 2021)7Zhu 2021 Dynamics of Convex Mean Curvature Flow (Sigurd Angenent, Panagiota Daskalopoulos and Natasa Sesum, 2023)8Angenent 2023 A Bernstein theorem of ancient solutions to mean curvature flow (Xiangzhi Cao, 2023)3Cao 2023 A Bernstein theorem for ancient solution to symplectic mean curvature flow (Xiangzhi Cao, 2023)5Cao 2023 A rigidity theorem of ancient solutions to the mean curvature flow in codimension one (Qun Chen and Hongbing Qiu, 2024)1Chen 2024 Planarity and convexity for pinched ancient solutions of mean curvature flow (Tang-Kai Lee, Keaton Naff and Jingze Zhu, 2025)2Lee 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A rigidity theorem of ancient solutions to the mean curvature flow in codimension one

    Qun Chen and Hongbing QiuWider flows PDF

    Topics Ancient solutions, Boundaries & non-compact, Stability & uniqueness · Same author Hongbing Qiu · Shared terms “rigidity theorem”, “noncompact ancient”, “carrying”

  2. Planarity and convexity for pinched ancient solutions of mean curvature flow

    Tang-Kai Lee, Keaton Naff and Jingze Zhu PDF

    Topics Ancient solutions · Shared terms “higher codimension”

  3. A Bernstein theorem of ancient solutions to mean curvature flow

    Xiangzhi CaoWider flows PDF

    Topics Ancient solutions

  4. Ancient solutions to the Ricci flow in dimension 3

    S. BrendleThe canon PDF

    Topics Ancient solutions, Boundaries & non-compact · Shared terms “noncompact ancient”

  5. A Bernstein theorem for ancient solution to symplectic mean curvature flow

    Xiangzhi CaoWider flows PDF

    Topics Ancient solutions

  6. A flow approach to the Musielak-Orlicz-Gauss image problem

    Qi-Rui Li, Weimin Sheng, Deping Ye et al.Wider flows PDF

    Shared terms “Gauss image”

  7. Quantitative long range curvature estimate for mean curvature flow

    Jingze ZhuWider flows PDF

    Topics Ancient solutions · Shared terms “curvature estimates”

  8. Dynamics of Convex Mean Curvature Flow

    Sigurd Angenent, Panagiota Daskalopoulos and Natasa SesumWider flows PDF

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

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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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.