Paper · Nov 12, 2022

Mean curvature flow solitons from symmetry group viewpoint

Xu Han, Zhonghua Hou

math.DGWider flowsarXiv:2211.06577v2
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Abstract

The symmetry group of the mean curvature flow in general ambient Riemannian manifolds is determined, based on which we define generalized solitons to the mean curvature flow. We also provide examples of homothetic solitons in non-Euclidean surfaces and prove that all the affine solutions to the mean curvature flow are self-similar solutions.

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Submitted
Nov 12, 2022
Last revised
Aug 4, 2023 (v2)
Subjects
math.DG, math.AP
Mean curvature flow

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The 5 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 2021, plus the canon.

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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. On mean curvature flow solitons in the sphere

    Marco Magliaro, Luciano Mari, Fernanda Roing et al.Wider flows

    Topics Mean curvature flow

  2. Self-shrinkers whose asymptotic cones fatten

    Daniel KetoverWider flows

    Topics Mean curvature flow · Shared terms “symmetry group”

  3. Classification of ruled surfaces as homothetic self-similar solutions of the inverse mean curvature flow in the Lorentz-Minkowski 3-space

    Gregório Silva Neto and Vanessa SilvaWider flows

    Topics Mean curvature flow · Shared terms “homothetic solitons”

  4. Mean curvature flow into an ambient Riemannian manifold evolving by Ricci flow coupled with harmonic map heat flow

    José N. V. Gomes, Matheus Hudson and Carlos M. de Sousa

    Topics Mean curvature flow · Shared terms “ambient Riemannian”, “Riemannian manifolds”

  5. Solitons of the mean curvature flow in

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    Topics Mean curvature flow

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. 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.