Paper · Nov 7, 2017

Gauss map of the skew mean curvature flow

Chong Song

math.DGWider flowsarXiv:1711.02312v2
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Abstract

The skew mean curvature flow (SMCF) is a natural generalization of the famous vortex filament equation. In this note, we show that the Gauss map of the SMCF satisfies a Schrödinger flow equation. In this regard, we explore the geometry of the oriented Grassmannian manifold explicitly by embedding it into the exterior product space.

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Submitted
Nov 7, 2017
Last revised
Nov 10, 2017 (v2)
Subjects
math.DG
Journal
Proc. Amer. Math. Soc. 145 (2017), no. 11, 4963-4970

Comments: Ref. [12] corrected

Mean curvature flow

Its neighbourhood

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Local existence and uniqueness of Skew Mean Curvature Flow

    Chong SongWider flows

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

  2. 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”

  3. Global and local theory of skew mean curvature flows

    Ze LiWider flows

    Topics Mean curvature flow · Shared terms “skew mean”

  4. Correspondence between Mean Curvature Flow and Harmonic-Ricci Flow

    Tianyin Ren and Chong Song

    Topics Mean curvature flow · Same author Chong Song · Shared terms “Gauss map”

  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”, “Schrödinger flow”

  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. Gauss maps of the Ricci-mean curvature flow

    Naoyuki Koike and Hikaru Yamamoto

    Topics Mean curvature flow · Shared terms “Gauss map”

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)

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.