Paper · Apr 8, 2019

Local existence and uniqueness of Skew Mean Curvature Flow

Chong Song

math.DGWider flowsarXiv:1904.03822
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Abstract

The Skew Mean Curvature Flow(SMCF) is a Schrödinger-type geometric flow canonically defined on a co-dimension two submanifold, which generalizes the famous vortex filament equation in fluid dynamics. In this paper, we prove the local existence and uniqueness of general dimensional SMCF in Euclidean spaces.

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Submitted
Apr 8, 2019
Subjects
math.DG
Mean curvature flowStability & uniqueness

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 Solitons of Curve Shortening Flow and Vortex Filament Equation (Bernardo Antonio Hernandez Adame, 2017)8Adame 2017 Gauss map of the skew mean curvature flow (Chong Song, 2017)1Song 2017 Global transversal stability of Euclidean planes under skew mean curvature flow evolutions (Ze Li, 2020)5Li 2020 Global and local theory of skew mean curvature flows (Ze Li, 2020)2Li 2020 Local well-posedness of skew mean curvature flow for small data in d≥ 4 dimensions (Jiaxi Huang and Daniel Tataru, 2021)3Huang 2021 Local well-posedness of the Skew mean curvature flow for small data in d≥ 2 dimensions (Jiaxi Huang and Daniel Tataru, 2022)7Huang 2022 Global regularity of Skew mean curvature flow for small data in d≥ 4 dimensions (Jiaxi Huang, Ze Li and Daniel Tataru, 2022)6Huang 2022 Local well-posedness of the skew mean curvature flow for large data (Jiaxi Huang and Daniel Tataru, 2025)4Huang 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Gauss map of the skew mean curvature flow

    Chong SongWider flows

    Topics Mean curvature flow · Same author Chong Song · Shared terms “skew mean”, “SMCF”, “famous vortex”

  2. Global and local theory of skew mean curvature flows

    Ze LiWider flows

    Topics Mean curvature flow · Shared terms “skew mean”

  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”, “Schrödinger”

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

    Jiaxi Huang and Daniel TataruWider flows

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

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

    Ze LiWider flows

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

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

    Jiaxi Huang, Ze Li and Daniel TataruWider flows

    Topics Mean curvature flow · Shared terms “skew mean”

  7. 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”, “Schrödinger”

  8. Solitons of Curve Shortening Flow and Vortex Filament Equation

    Bernardo Antonio Hernandez AdameWider flows

    Shared terms “filament equation”, “vortex filament”

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) 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)

In the guidebook

  1. 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.