Paper · Mar 7, 2018

Translating solitons of the mean curvature flow in the space

Antonio Bueno

math.DGWider flowsarXiv:1803.02783v3
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Abstract

In this paper we study the theory of self translating solitons of the mean curvature flow of immersed surfaces in the product space . We relate this theory to the one of manifolds with density, and exploit this relation by regarding these translating solitons as minimal surfaces in a conformal metric space. Explicit examples of these surfaces are constructed, and we study the asymptotic behaviour of the existing rotationally symmetric examples. Finally, we prove some uniqueness and non-existence theorems.

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Submitted
Mar 7, 2018
Last revised
Aug 19, 2018 (v3)
Subjects
math.DG

Comments: 26 pages, 11 figures

Mean curvature flowSelf-shrinkers & translatorsStability & uniquenessRotational symmetry & warped productsMinimal & CMC surfaces

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 The global geometry of surfaces with prescribed mean curvature in ℝ³ (Antonio Bueno, Jose A. Galvez and Pablo Mira, 2018)1Bueno 2018 Translating solitons in Riemannian products (Jorge H. de Lira and Francisco Martin, 2018)3de Lira 2018 The Björling problem for prescribed mean curvature surfaces in ℝ³ (Antonio Bueno, 2018)4Bueno 2018 Bi-Halfspace and Convex Hull Theorems for Translating Solitons (Francesco Chini and Niels Martin Møller, 2018)6Chini 2018 Surfaces of prescribed linear Weingarten curvature in ℝ³ (Antonio Bueno and Irene Ortiz, 2022)7Bueno 2022 Finite entropy translating solitons in slabs (Eddygledson Souza Gama, Francisco Martín and Niels Martin Møller, 2022)5Gama 2022 Dynamical Stability of Translating Solitons to Mean Curvature Flow in Hyperbolic Space (Ronaldo F. de Lima and Álvaro K. Ramos, 2026)8de Lima 2026 Invariant λ-translators in the Heisenberg group (Antonio Bueno, 2026)2Bueno 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The global geometry of surfaces with prescribed mean curvature in

    Antonio Bueno, Jose A. Galvez and Pablo MiraWider flows

    Topics Minimal & CMC surfaces, Self-shrinkers & translators · Same author Antonio Bueno · Shared terms “self translating”, “translating solitons”, “regarding”

  2. Invariant -translators in the Heisenberg group

    Antonio BuenoWider flows

    Topics Minimal & CMC surfaces, Self-shrinkers & translators · Same author Antonio Bueno · Shared terms “translating solitons”

  3. Translating solitons in Riemannian products

    Jorge H. de Lira and Francisco MartinWider flows

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

  4. The Björling problem for prescribed mean curvature surfaces in

    Antonio BuenoWider flows

    Topics Self-shrinkers & translators · Same author Antonio Bueno · Shared terms “immersed surface”, “self translating”, “translating solitons”

  5. Finite entropy translating solitons in slabs

    Eddygledson Souza Gama, Francisco Martín and Niels Martin MøllerWider flows

    Topics Minimal & CMC surfaces, Self-shrinkers & translators, Stability & uniqueness · Shared terms “translating solitons”

  6. Bi-Halfspace and Convex Hull Theorems for Translating Solitons

    Francesco Chini and Niels Martin MøllerWider flows

    Topics Minimal & CMC surfaces, Self-shrinkers & translators · Shared terms “translating solitons”, “self translating”

  7. Surfaces of prescribed linear Weingarten curvature in

    Antonio Bueno and Irene OrtizWider flows

    Topics Self-shrinkers & translators · Same author Antonio Bueno · Shared terms “self translating”, “translating solitons”

  8. 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, Stability & uniqueness · 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) 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) 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.