Paper · Jun 6, 2023

Maximum Principles and Consequences for -translators in

José Torres Santaella

math.DGWider flowsarXiv:2306.03649
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Abstract

In this paper we obtain several properties of translating solitons for a general class of extrinsic geometric curvature flows given by a homogeneous, symmetric, smooth non-negative function defined in an open cone . The main results are tangential principles, nonexistence theorems for closed and entire solutions, and a uniqueness result that says that any strictly convex -translator defined on a ball with a single end -asymptotic to a cylinder is the "bowl"-type solution found in the translator paper of S. Rengaswami.

Comments: Comments are welcome!

Other curvature flowsSelf-shrinkers & translatorsHomogeneous spacesStability & uniqueness

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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 Rotationally symmetric translating solutions to extrinsic geometric flows (Sathyanarayanan Rengaswami, 2021)3Rengaswami 2021 Solitons to Mean Curvature Flow in the hyperbolic 3-space (R. F. de Lima, A. K. Ramos and J. P. dos Santos, 2023)7de Lima 2023 Translators invariant under hyperpolar actions (Tomoki Fujii and Naoyuki Koike, 2023)4Fujii 2023 Bowl Soliton Asymptotics and Applications (Sathya Rengaswami and José Torres Santaella, 2023)2Rengaswami 2023 Classification of Semigraphical Translators (Francisco Martín, Mariel Sáez, Raphael Tsiamis et al., 2024)6Martín 2024 Rotationally symmetric translating solitons of fully nonlinear extrinsic geometric flows: Classification and Applications (José Torres Santaella, 2025)1Santaella 2025 Dynamical Stability of Translating Solitons to Mean Curvature Flow in Hyperbolic Space (Ronaldo F. de Lima and Álvaro K. Ramos, 2026)8de Lima 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 solitons of fully nonlinear extrinsic geometric flows: Classification and Applications

    José Torres SantaellaWider flows PDF

    Topics Self-shrinkers & translators, Other curvature flows, Stability & uniqueness · Same author José Torres Santaella · Shared terms “extrinsic geometric”, “translating solitons”, “nonexistence”

  2. Bowl Soliton Asymptotics and Applications

    Sathya Rengaswami and José Torres SantaellaWider flows PDF

    Topics Self-shrinkers & translators, Other curvature flows, Stability & uniqueness · Same author José Torres Santaella

  3. Rotationally symmetric translating solutions to extrinsic geometric flows

    Sathyanarayanan RengaswamiWider flows PDF

    Topics Self-shrinkers & translators, Homogeneous spaces, Other curvature flows · Shared terms “extrinsic geometric”

  4. Translators invariant under hyperpolar actions

    Tomoki Fujii and Naoyuki KoikeWider flows PDF

    Topics Self-shrinkers & translators, Homogeneous spaces

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

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

    Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “maximum principle”, “translating solitons”

  6. Classification of Semigraphical Translators

    Francisco Martín, Mariel Sáez, Raphael Tsiamis et al.Wider flows PDF

    Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “maximum principle”

  7. 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, Stability & uniqueness · Shared terms “translating solitons”

  8. 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, Stability & uniqueness · Shared terms “translating solitons”, “maximum principle”

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.

Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) 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) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

In the guidebook

  1. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.