Paper · Jan 27, 2016

Some results on translating solitons of the mean curvature flow

Jesús Pérez-García

math.DGWider flowsarXiv:1601.07287
View PDF View HTML arXiv page

As on arXiv

Abstract

In this article we prove two non-existence results for translating solitons of the mean curvature flow (translators for short) in . We also obtain an upper bound to the maximum height that a compact embedded translator in can achieve. On the other hand, we study graphical perturbations of translators, showing that asymptotic graphical perturbations of a graph translator of revolution remain a hypersurface of revolution. Finally, we prove that compact translators that lie between two parallel planes inherit the symmetries of their boundaries curves.

View PDF View HTML arXiv page
Submitted
Jan 27, 2016
Subjects
math.DG

Comments: 20 pages, 3 figures

Mean curvature flowSelf-shrinkers & translatorsRotational symmetry & warped products

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 Simply connected translating solitons contained in slabs (Francesco Chini, 2019)8Chini 2019 SO(2) symmetry of the translating solitons of the mean curvature flow in ℝ⁴ (Jingze Zhu, 2020)3Zhu 2020 Rotational symmetry of uniformly 3-convex translating solitons of mean curvature flow in higher dimensions (Jingze Zhu, 2021)1Zhu 2021 Translators of the Mean Curvature Flow in Hyperbolic Einstein's Static Universe (Miguel Ortega and Buse Yalçın, 2024)5Ortega 2024 Rotationally symmetric translating solitons of fully nonlinear extrinsic geometric flows: Classification and Applications (José Torres Santaella, 2025)6Santaella 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 Translating solitons to higher order mean curvature flows in Riemannian products (Jorge Herbert Soares de Lira and Rafael Rocha de Farias, 2026)7de Lira 2026 A Bernstein problem for translating solutions to the mean curvature flow (Juan Davila, Manuel del Pino and Juncheng Wei, 2026)4Davila 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Rotational symmetry of uniformly 3-convex translating solitons of mean curvature flow in higher dimensions

    Jingze ZhuWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators · Shared terms “translating solitons”

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

    Ronaldo F. de Lima and Álvaro K. RamosWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators · Shared terms “translating solitons”, “graphical”

  3. symmetry of the translating solitons of the mean curvature flow in

    Jingze ZhuWider flows

    Topics Self-shrinkers & translators · Shared terms “translating solitons”

  4. A Bernstein problem for translating solutions to the mean curvature flow

    Juan Davila, Manuel del Pino and Juncheng WeiWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators · Shared terms “graphical”, “translating solitons”, “perturbation”

  5. Translators of the Mean Curvature Flow in Hyperbolic Einstein's Static Universe

    Miguel Ortega and Buse YalçınWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators

  6. Rotationally symmetric translating solitons of fully nonlinear extrinsic geometric flows: Classification and Applications

    José Torres SantaellaWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators · Shared terms “translating solitons”, “graphical”

  7. Translating solitons to higher order mean curvature flows in Riemannian products

    Jorge Herbert Soares de Lira and Rafael Rocha de FariasWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators · Shared terms “translating solitons”

  8. Simply connected translating solitons contained in slabs

    Francesco ChiniWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators · Shared terms “translating solitons”

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) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

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.