Paper · Dec 13, 2020

The Structure of Translating Surfaces with Finite Total Curvature

Ilyas Khan

math.DGWider flowsarXiv:2012.06926v2
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Abstract

In this paper, we prove that any mean curvature flow translator with finite total curvature and one end must be a plane. We also prove that if the translator has multiple ends, they are asymptotic to a plane containing the direction of translation and can be written as graphs over . Finally, we determine that the ends of are strongly asymptotic to and obtain quantitative estimates for their asymptotic behavior.

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Submitted
Dec 13, 2020
Last revised
Jun 21, 2021 (v2)
Subjects
math.DG

Comments: 26 Pages, significantly expanded: new results include asymptotic estimates and removal of conditions on area ratios. More detailed exposition, includes 2 new figures

Mean curvature flowSelf-shrinkers & translators

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 Some results on translating solitons of the mean curvature flow (Jesús Pérez-García, 2016)8Pérez-García 2016 Translating surfaces of the non-parametric mean curvature flow in Lorentz manifold M²×ℝ (Li Chen, Dan-Dan Hu, Jing Mao et al., 2018)5Chen 2018 Uniqueness of Asymptotically Conical Higher Codimension Self-Shrinkers and Self-Expanders (Ilyas Khan, 2020)3Khan 2020 An unknottedness result for self shrinkers with multiple ends (Alexander Mramor, 2020)4Mramor 2020 Nonconvex ancient solutions to Curve Shortening Flow (Yongzhe Zhang, Connor Olson, Ilyas Khan et al., 2022)7Zhang 2022 Rigidity results for mean curvature flow graphical translators moving in non-graphical direction (John Man Shun Ma, Yuan Shyong Ooi and Juncheol Pyo, 2022)2Ma 2022 On mean curvature flow translators with prescribed ends (Ao Sun and Zhihan Wang, 2023)6Sun 2023 Ancient mean curvature flows with finite total curvature (Kyeongsu Choi, Jiuzhou Huang and Taehun Lee, 2024)1Choi 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ancient mean curvature flows with finite total curvature

    Kyeongsu Choi, Jiuzhou Huang and Taehun LeeWider flows

    Topics Mean curvature flow · Shared terms “finite total”, “total curvature”

  2. Rigidity results for mean curvature flow graphical translators moving in non-graphical direction

    John Man Shun Ma, Yuan Shyong Ooi and Juncheol PyoWider flows

    Topics Self-shrinkers & translators · Shared terms “translating surface”, “direction”

  3. Uniqueness of Asymptotically Conical Higher Codimension Self-Shrinkers and Self-Expanders

    Ilyas KhanWider flows

    Topics Self-shrinkers & translators · Same author Ilyas Khan

  4. An unknottedness result for self shrinkers with multiple ends

    Alexander MramorWider flows

    Topics Self-shrinkers & translators · Shared terms “multiple ends”

  5. Translating surfaces of the non-parametric mean curvature flow in Lorentz manifold

    Li Chen, Dan-Dan Hu, Jing Mao et al.Wider flows

    Topics Self-shrinkers & translators · Shared terms “translating surface”, “translations”

  6. On mean curvature flow translators with prescribed ends

    Ao Sun and Zhihan WangWider flows

    Topics Self-shrinkers & translators

  7. Nonconvex ancient solutions to Curve Shortening Flow

    Yongzhe Zhang, Connor Olson, Ilyas Khan et al.Wider flows

    Topics Self-shrinkers & translators · Same author Ilyas Khan

  8. Some results on translating solitons of the mean curvature flow

    Jesús Pérez-GarcíaWider flows

    Topics Self-shrinkers & translators

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) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990)

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.