Paper · Jan 2, 2021

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

Jiaxi Huang, Daniel Tataru

math.APWider flowsarXiv:2101.00358v2
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Abstract

The skew mean curvature flow is an evolution equation for dimensional manifolds embedded in (or more generally, in a Riemannian manifold). It can be viewed as a Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger Map equation. In this article, we prove small data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension .

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Submitted
Jan 2, 2021
Last revised
Jul 23, 2021 (v2)
Subjects
math.AP
DOI
10.1007/s00220-021-04303-8
Mean curvature flow

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    Jiaxi Huang and Daniel TataruWider flows

    Topics Mean curvature flow · Same authors Jiaxi Huang, Daniel Tataru · Shared terms “small data”, “skew mean”, “posedness”

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

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    Topics Mean curvature flow · Same authors Jiaxi Huang, Daniel Tataru · Shared terms “small data”, “skew mean”, “regularity Sobolev”

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

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

  6. Local existence and uniqueness of Skew Mean Curvature Flow

    Chong SongWider flows

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

  7. Gauss map of the skew mean curvature flow

    Chong SongWider flows

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

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

    Ze LiWider flows

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

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.