Paper · Jul 14, 2020

Global transversal stability of Euclidean planes under skew mean curvature flow evolutions

Ze Li

math.APWider flowsarXiv:2007.07007v3
View PDF View HTML arXiv page

As on arXiv

Abstract

In this paper, we prove that 2 dimensional transversal small perturbations of d-dimensional Euclidean planes under the skew mean curvature flow lead to global solutions which converge to the unperturbed planes in suitable norms. And we clarify the long time behaviors of the solutions in Sobolev spaces.

View PDF View HTML arXiv page
Submitted
Jul 14, 2020
Last revised
Dec 30, 2020 (v3)
Subjects
math.AP

Comments: Revised version. We acknowledge the reviewers for their insightful comments

Mean curvature flowSurfacesStability & uniquenessLong-time existence & convergence

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 Evolution of area-decreasing maps between two-dimensional Euclidean spaces (Felix Lubbe, 2016)3Lubbe 2016 Local existence and uniqueness of Skew Mean Curvature Flow (Chong Song, 2019)5Song 2019 Global and local theory of skew mean curvature flows (Ze Li, 2020)1Li 2020 Local well-posedness of skew mean curvature flow for small data in d≥ 4 dimensions (Jiaxi Huang and Daniel Tataru, 2021)7Huang 2021 Global regularity of Skew mean curvature flow for small data in d≥ 4 dimensions (Jiaxi Huang, Ze Li and Daniel Tataru, 2022)2Huang 2022 Global well-posedness in the critical Besov space of the skew mean curvature flow in ℝ^d: d≥ 5 (Ning-An Lai, Jie Shao, Zexian Zhang et al., 2025)8Lai 2025 Dynamical stability of various convex graphical translators (Junyoung Park, 2026)4Park 2026 Calibration energy and mean curvature flow (Tatsuya Miura and Fabian Rupp, 2026)6Miura 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Global and local theory of skew mean curvature flows

    Ze LiWider flows

    Topics Mean curvature flow · Same author Ze Li · Shared terms “skew mean”, “Sobolev spaces”, “small perturbations”

  2. Global regularity of Skew mean curvature flow for small data in dimensions

    Jiaxi Huang, Ze Li and Daniel TataruWider flows

    Topics Mean curvature flow · Same author Ze Li · Shared terms “skew mean”, “Sobolev spaces”

  3. Evolution of area-decreasing maps between two-dimensional Euclidean spaces

    Felix LubbeWider flows

    Topics Surfaces, Long-time existence & convergence · Shared terms “dimensional Euclidean”

  4. Dynamical stability of various convex graphical translators

    Junyoung ParkWider flows

    Topics Surfaces, Long-time existence & convergence, Stability & uniqueness

  5. Local existence and uniqueness of Skew Mean Curvature Flow

    Chong SongWider flows

    Topics Stability & uniqueness · Shared terms “skew mean”, “Euclidean”

  6. Calibration energy and mean curvature flow

    Tatsuya Miura and Fabian RuppWider flows

    Topics Surfaces, Long-time existence & convergence, Stability & uniqueness · Shared terms “Euclidean”

  7. Local well-posedness of skew mean curvature flow for small data in dimensions

    Jiaxi Huang and Daniel TataruWider flows

    Topics Mean curvature flow · Shared terms “skew mean”, “Sobolev spaces”

  8. Global well-posedness in the critical Besov space of the skew mean curvature flow in

    Ning-An Lai, Jie Shao, Zexian Zhang et al.Wider flows

    Topics Long-time existence & convergence · Shared terms “skew mean”, “global solution”

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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988) 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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.