Paper · Oct 27, 2020

Global and local theory of skew mean curvature flows

Ze Li

math.APWider flowsarXiv:2010.14011v2
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Abstract

In this paper, we study the skew mean curvature flow. The results are threefold. First, we prove the global regularity of solutions with initial data which are small perturbations of planes in Sobolev spaces. Second, we prove the modified scattering and the existence of wave operators for small data, which completely determines the set of asymptotic states. Third, we study the Cauchy problem for arbitrary large data.

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Submitted
Oct 27, 2020
Last revised
Nov 29, 2021 (v2)
Subjects
math.AP

Comments: The existence of wave operators was added. To appear in J. Geom. Anal

Mean curvature flow3-manifolds

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  1. 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”, “global regularity”, “small data”

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

    Ze LiWider flows

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

  3. 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”, “small data”, “Sobolev spaces”

  4. Local well-posedness of the Skew mean curvature flow for small data in dimensions

    Jiaxi Huang and Daniel TataruWider flows

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

  5. Local well-posedness of the skew mean curvature flow for large data

    Jiaxi Huang and Daniel TataruWider flows

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

  6. Local existence and uniqueness of Skew Mean Curvature Flow

    Chong SongWider flows

    Topics Mean curvature flow · Shared terms “skew mean”

  7. 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 Mean curvature flow · Shared terms “skew mean”, “small data”, “global regularity”

  8. Gauss map of the skew mean curvature flow

    Chong SongWider flows

    Topics Mean curvature flow · Shared terms “skew mean”

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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 10A.1
    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

  2. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  3. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.