Paper · Nov 27, 2018

Remarks on mean curvature flow solitons in warped products

Giulio Colombo, Luciano Mari, Marco Rigoli

math.DGWider flowsarXiv:1811.10968v4
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Abstract

We study some properties of mean curvature flow solitons in general Riemannian manifolds and in warped products, with emphasis on constant curvature and Schwarzschild type spaces. We focus on splitting and rigidity results under various geometric conditions, ranging from the stability of the soliton to the fact that the image of its Gauss map be contained in suitable regions of the sphere. We also investigate the case of entire graphs.

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Submitted
Nov 27, 2018
Last revised
Nov 21, 2021 (v4)
Subjects
math.DG
Journal
Discrete Contin. Dyn. Syst. Ser. S 13 (2020), no. 7, 1957-1991. Erratum on: Corrigendum on: DCDS - series S, v.16 no.1, p. 187-196, 2023
DOI
10.3934/dcdss.2020153

Comments: 36 pages. This paper differs from the published one since the proof of Theorem 3.4 (Thm. C in the Introduction) was rewritten. Package axessibility added to make the paper available to visually impaired people. Last version: some further detail added in the proof of Theorem 3.4

Mean curvature flowStability & uniquenessRotational symmetry & warped products

Its neighbourhood

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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 Inverse mean curvature flows in warped product manifolds (Hengyu Zhou, 2016)7Zhou 2016 On the inverse mean curvature flow in warped product manifolds (Thomas Mullins, 2016)5Mullins 2016 Mean curvature flow solitons in the presence of conformal vector fields (Luis Alías, Jorge H. de Lira and Marco Rigoli, 2017)2Alías 2017 Mean Curvature Flows of Closed Hypersurfaces in Warped Product Manifolds (Zheng Huang, Zhou Zhang and Hengyu Zhou, 2017)6Huang 2017 Self-similar solutions of curvature flows in warped products (Shanze Gao and Hui Ma, 2018)3Gao 2018 Recent rigidity results for graphs with prescribed mean curvature (Bruno Bianchini, Giulio Colombo, Marco Magliaro et al., 2020)1Bianchini 2020 Uniqueness of entire graphs evolving by Mean Curvature flow (Panagiota Daskalopoulos and Mariel Saez, 2021)8Daskalopoulo 2021 On mean curvature flow solitons in the sphere (Marco Magliaro, Luciano Mari, Fernanda Roing et al., 2025)4Magliaro 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Recent rigidity results for graphs with prescribed mean curvature

    Bruno Bianchini, Giulio Colombo, Marco Magliaro et al.Wider flows

    Topics Rotational symmetry & warped products, Stability & uniqueness · Same authors Giulio Colombo, Luciano Mari, Marco Rigoli · Shared terms “emphasis”, “splitting”

  2. 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 Marco Rigoli

  3. Self-similar solutions of curvature flows in warped products

    Shanze Gao and Hui MaWider flows

    Topics Rotational symmetry & warped products, Stability & uniqueness · Shared terms “Gauss”

  4. On mean curvature flow solitons in the sphere

    Marco Magliaro, Luciano Mari, Fernanda Roing et al.Wider flows

    Topics Rotational symmetry & warped products · Same author Luciano Mari

  5. On the inverse mean curvature flow in warped product manifolds

    Thomas MullinsWider flows

    Topics Rotational symmetry & warped products

  6. Mean Curvature Flows of Closed Hypersurfaces in Warped Product Manifolds

    Zheng Huang, Zhou Zhang and Hengyu ZhouWider flows

    Topics Rotational symmetry & warped products

  7. Inverse mean curvature flows in warped product manifolds

    Hengyu ZhouWider flows

    Topics Rotational symmetry & warped products

  8. Uniqueness of entire graphs evolving by Mean Curvature flow

    Panagiota Daskalopoulos and Mariel SaezWider flows

    Topics Rotational symmetry & warped products, Stability & uniqueness · Shared terms “entire graph”

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) 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.