Paper · Apr 26, 2019

A family of MCF solutions for the Heisenberg Group

Benedito Leandro, Adriana Araujo Cintra, Hiuri Fellipe dos Santos Reis

math.DGWider flowsarXiv:1904.12015

As on arXiv

Abstract

The aim of this paper is to investigate the mean curvature flow soliton solutions on the Heisenberg group when the initial data is a ruled surface by straight lines. We give a family of those solutions which are generated by (the isometries of for which the origin is a fix point). We conclude that the function which describe the motion of these surfaces under MCF, is always a linear affine function. As an application we proof that the Grim Reaper solution evolves from a ruled surface in . We also provide other examples.

Submitted
Apr 26, 2019
Subjects
math.DG
Journal
Differential Geometry and its Applications 2020
DOI
10.1016/j.difgeo.2020.101633

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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 Invariant translators of the Heisenberg group (Giuseppe Pipoli, 2018)3Pipoli 2018 Ruled surfaces of generalized self-similar solutions of the mean curvature flow (Rafael López, 2020)2López 2020 Self-similar solutions to the mean curvature flow in ℝ³ (Benedito Leandro, Rafael Novais and Hiuri F. S. dos Reis, 2020)1Leandro 2020 Translators to Higher Order Mean Curvature Flows in ℝⁿ×ℝ and ℍⁿ×ℝ (Ronaldo F. de Lima and Giuseppe Pipoli, 2022)7de Lima 2022 Solitons to Mean Curvature Flow in the hyperbolic 3-space (R. F. de Lima, A. K. Ramos and J. P. dos Santos, 2023)8de Lima 2023 Ruled surfaces as translating solitons of the inverse mean curvature flow in the three-dimensional Lorentz-Minkowski space (Gregório Silva Neto and Vanessa Silva, 2023)5Neto 2023 The class of grim reapers in ℍ²×ℝ (Antonio Bueno and Rafael López, 2024)6Bueno 2024 Invariant λ-translators in the Heisenberg group (Antonio Bueno, 2026)4Bueno 2026
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  1. Self-similar solutions to the mean curvature flow in

    Benedito Leandro, Rafael Novais and Hiuri F. S. dos ReisWider flows

    Topics Mean curvature flow · Same author Benedito Leandro · Shared terms “MCF”, “ruled surfaces”

  2. Ruled surfaces of generalized self-similar solutions of the mean curvature flow

    Rafael LópezWider flows

    Topics Self-shrinkers & translators · Shared terms “ruled surfaces”

  3. Invariant translators of the Heisenberg group

    Giuseppe PipoliWider flows

    Topics Self-shrinkers & translators · Shared terms “Heisenberg group”, “grim reaper”, “isometry”

  4. Invariant -translators in the Heisenberg group

    Antonio BuenoWider flows

    Topics Self-shrinkers & translators · Shared terms “Heisenberg group”, “isometry”

  5. The class of grim reapers in

    Antonio Bueno and Rafael LópezWider flows

    Topics Self-shrinkers & translators · Shared terms “grim reaper”

  6. Translators to Higher Order Mean Curvature Flows in and

    Ronaldo F. de Lima and Giuseppe PipoliWider flows

    Topics Self-shrinkers & translators · Shared terms “MCF”, “grim reaper”

  7. Solitons to Mean Curvature Flow in the hyperbolic 3-space

    R. F. de Lima, A. K. Ramos and J. P. dos SantosWider flows

    Topics Self-shrinkers & translators · Shared terms “MCF”

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Background reading

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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.