Paper · May 22, 2025

Existence proofs for rotationally symmetric translating solutions to mean curvature flow

Hakar Raji, Oliver C. Schnürer

math.DGWider flowsarXiv:2505.16688
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Abstract

There exist rotationally symmetric translating solutions to mean curvature flow that can be written as a graph over Euclidean space. This result is well-known. Its proof uses the symmetry and techniques from partial differential equations. However, the result can also be formulated as an existence result for a singular ordinary differential equation. Here, we provide different methods to prove existence of these solutions based on the study of the singular ordinary differential equation without using methods from partial differential equations.

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Submitted
May 22, 2025
Subjects
math.DG

Comments: 35 pages, 3 figures, 2 sniplets of python code

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 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 Rotationally symmetric translating solutions to extrinsic geometric flows (Sathyanarayanan Rengaswami, 2021)1Rengaswami 2021 Graphical translating solitons for the mean curvature flow and isoparametric functions (Tomoki Fujii, 2021)8Fujii 2021 Translating Solutions of a Generalized Mean Curvature Flow in a Cylinder: I. Constant Boundary Angles (Bendong Lou and Lixia Yuan, 2022)2Lou 2022 Rotationally symmetric translating solitons of fully nonlinear extrinsic geometric flows: Classification and Applications (José Torres Santaella, 2025)5Santaella 2025 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 Comparison Theorems for the Profile Curve Equation of Rotationally Symmetric Self-Shrinkers (Peng Peng, 2026)3Peng 2026 A Bernstein problem for translating solutions to the mean curvature flow (Juan Davila, Manuel del Pino and Juncheng Wei, 2026)4Davila 2026 Instability of mean curvature flow in expander coordinates (Ben S. Lambert and Oliver C. Schnürer, 2026)6Lambert 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Rotationally symmetric translating solutions to extrinsic geometric flows

    Sathyanarayanan RengaswamiWider flows

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

  2. Translating Solutions of a Generalized Mean Curvature Flow in a Cylinder: I. Constant Boundary Angles

    Bendong Lou and Lixia YuanWider flows

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

  3. Comparison Theorems for the Profile Curve Equation of Rotationally Symmetric Self-Shrinkers

    Peng PengWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators · Shared terms “rotationally symmetric”, “differential equation”, “ordinary differential”

  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 “translating solution”, “rotationally symmetric”

  5. 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 “symmetric translating”, “rotationally symmetric”, “Euclidean space”

  6. Instability of mean curvature flow in expander coordinates

    Ben S. Lambert and Oliver C. Schnürer

    Topics Rotational symmetry & warped products · Same author Oliver C. Schnürer · Shared terms “rotationally symmetric”

  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 “rotationally symmetric”

  8. Graphical translating solitons for the mean curvature flow and isoparametric functions

    Tomoki FujiiWider flows

    Topics Self-shrinkers & translators · Shared terms “differential equation”, “ordinary differential”

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) 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.