Paper · Mar 4, 2018

Translating solitons in Riemannian products

Jorge H. de Lira, Francisco Martin

math.DGWider flowsarXiv:1803.01410
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Abstract

In this paper we study solitons invariant with respect to the flow generated by a complete Killing vector field in a ambient Riemannian manifold. A special case occurs when the ambient manifold is the Riemannian product and the Killing field is . Similarly to what happens in the Euclidean setting, we call them translating solitons. We see that a translating soliton in can be seen as a minimal submanifold for a weighted volume functional. Moreover we show that this kind of solitons appear in a natural way in the context of a monotonicity formula for the mean curvature flow in . When is rotationally invariant and its sectional curvature is non-positive, we are able to characterize all the rotationally invariant translating solitons. Furthermore, we use these families of new examples as barriers to deduce several non-existence results.

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Submitted
Mar 4, 2018
Subjects
math.DG

Comments: 32 pages, 1 figure

Mean curvature flowSelf-shrinkers & translatorsRotational 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 Mean curvature flow solitons in the presence of conformal vector fields (Luis Alías, Jorge H. de Lira and Marco Rigoli, 2017)6Alías 2017 Translating solitons of the mean curvature flow in the space ℍ²×ℝ (Antonio Bueno, 2018)4Bueno 2018 Jenkins-Serrin problem for translating horizontal graphs in M ×ℝ (Eddygledson S. Gama, Esko Heinonen, Jorge H. de Lira et al., 2019)2Gama 2019 Notes on translating solitons for Mean Curvature Flow (David Hoffman, Tom Ilmanen, Francisco Martín et al., 2019)8Hoffman 2019 Scherk-like Translators for Mean Curvature Flow (David Hoffman, Francisco Martín and Brian White, 2019)7Hoffman 2019 Non-parametric mean curvature flow with prescribed contact angle in Riemannian products (Jean-Baptiste Casteras, Esko Heinonen, Ilkka Holopainen et al., 2020)5Casteras 2020 Finite entropy translating solitons in slabs (Eddygledson Souza Gama, Francisco Martín and Niels Martin Møller, 2022)3Gama 2022 Translating solitons to higher order mean curvature flows in Riemannian products (Jorge Herbert Soares de Lira and Rafael Rocha de Farias, 2026)1de Lira 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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 · Same author Jorge H. de Lira · Shared terms “Riemannian product”, “translating solitons”

  2. Jenkins-Serrin problem for translating horizontal graphs in

    Eddygledson S. Gama, Esko Heinonen, Jorge H. de Lira et al.Wider flows

    Topics Self-shrinkers & translators · Same authors Jorge H. de Lira, Francisco Martin · Shared terms “Riemannian product”, “translating solitons”

  3. Finite entropy translating solitons in slabs

    Eddygledson Souza Gama, Francisco Martín and Niels Martin MøllerWider flows

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

  4. Translating solitons of the mean curvature flow in the space

    Antonio BuenoWider flows

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

  5. Non-parametric mean curvature flow with prescribed contact angle in Riemannian products

    Jean-Baptiste Casteras, Esko Heinonen, Ilkka Holopainen et al.Wider flows

    Topics Self-shrinkers & translators · Same author Jorge H. de Lira · Shared terms “Riemannian product”, “translating solitons”, “Euclidean setting”

  6. Mean curvature flow solitons in the presence of conformal vector fields

    Luis Alías, Jorge H. de Lira and Marco RigoliWider flows

    Topics Rotational symmetry & warped products · Same author Jorge H. de Lira · Shared terms “Riemannian product”, “vector field”, “sectional curvature”

  7. Scherk-like Translators for Mean Curvature Flow

    David Hoffman, Francisco Martín and Brian WhiteWider flows

    Topics Minimal & CMC surfaces, Self-shrinkers & translators · Same author Francisco Martin

  8. Notes on translating solitons for Mean Curvature Flow

    David Hoffman, Tom Ilmanen, Francisco Martín et al.Wider flows

    Topics Self-shrinkers & translators · Same author Francisco Martin · 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) Minimal & CMC surfaces Minimal surfaces are critical points of area, with zero mean curvature; constant mean curvature (CMC) surfaces are critical for area at fixed volume. Start with: On the structure of manifolds with positive scalar curvature (Schoen et al., 1979)

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.