Paper · Feb 7, 2024

Rotators-Translators to Mean Curvature Flow in

Ronaldo F. de Lima, Álvaro K. Ramos, João Paulo dos Santos

math.DGWider flowsarXiv:2402.04757
PDF Abstract page

As on arXiv

Abstract

We establish the existence of one-parameter families of helicoidal surfaces of which, under mean curvature flow, simultaneously rotate about a vertical axis and translate vertically.

Comments: 10 pages, 4 figures. arXiv admin note: text overlap with arXiv:2307.14136

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 2021, 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 Uniqueness of the [φ,e₃]-catenary cylinders by their asymptotic behaviour (A. L. Martínez-Triviño and J. P. dos Santos, 2021)5Martínez-Tri 2021 Translators to Higher Order Mean Curvature Flows in ℝⁿ×ℝ and ℍⁿ×ℝ (Ronaldo F. de Lima and Giuseppe Pipoli, 2022)3de 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)1de Lima 2023 Translating Annuli for Mean Curvature Flow (David Hoffman, Francisco Martín and Brian White, 2023)6Hoffman 2023 Hessian estimates for shrinkers, expanders, translators, and rotators of the Lagrangian Mean Curvature Flow (Arunima Bhattacharya and Jeremy Wall, 2024)4Bhattacharya 2024 Classification of Minimal Surfaces and Solitons to the Mean Curvature Flow in ℍ³ as Translation Surfaces (Tarcios Andrey Ferreira and João Paulo dos Santos, 2025)7Ferreira 2025 Dynamical Stability of Translating Solitons to Mean Curvature Flow in Hyperbolic Space (Ronaldo F. de Lima and Álvaro K. Ramos, 2026)2de Lima 2026 Invariant λ-translators in the Heisenberg group (Antonio Bueno, 2026)8Bueno 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Solitons to Mean Curvature Flow in the hyperbolic 3-space

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

    Topics Self-shrinkers & translators · Same authors Ronaldo F. de Lima, Álvaro K. Ramos, João Paulo dos Santos · Shared terms “parameter family”, “helicoidal”

  2. Dynamical Stability of Translating Solitons to Mean Curvature Flow in Hyperbolic Space

    Ronaldo F. de Lima and Álvaro K. RamosWider flows PDF

    Topics Self-shrinkers & translators · Same authors Ronaldo F. de Lima, Álvaro K. Ramos

  3. Translators to Higher Order Mean Curvature Flows in and

    Ronaldo F. de Lima and Giuseppe PipoliWider flows PDF

    Topics Self-shrinkers & translators · Same author Ronaldo F. de Lima

  4. Uniqueness of the -catenary cylinders by their asymptotic behaviour

    A. L. Martínez-Triviño and J. P. dos SantosWider flows PDF

    Topics Self-shrinkers & translators · Same author João Paulo dos Santos · Shared terms “vertical”

  5. Translating Annuli for Mean Curvature Flow

    David Hoffman, Francisco Martín and Brian WhiteWider flows PDF

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

  6. Invariant -translators in the Heisenberg group

    Antonio BuenoWider flows PDF

    Topics Self-shrinkers & translators · Shared terms “helicoidal”, “vertical”

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.

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.