Paper · Jul 22, 2017

Mean curvature flow solitons in the presence of conformal vector fields

Luis Alías, Jorge H. de Lira, Marco Rigoli

math.DGWider flowsarXiv:1707.07132v2
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Abstract

The aim of this paper is to introduce a notion of mean curvature flow soliton general enough to encompass target spaces of constant sectional curvature, Riemannian products or, in increasing generality, warped product spaces.

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Submitted
Jul 22, 2017
Last revised
Jul 25, 2017 (v2)
Subjects
math.DG

Comments: Some minor corrections in references

Mean curvature flowRotational 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 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 Long Time Existence of IMCF on Metrics Conformal to Warped Product Manifolds (Brian Allen, 2017)4Allen 2017 Translating solitons in Riemannian products (Jorge H. de Lira and Francisco Martin, 2018)3de Lira 2018 Remarks on mean curvature flow solitons in warped products (Giulio Colombo, Luciano Mari and Marco Rigoli, 2018)1Colombo 2018 Recent rigidity results for graphs with prescribed mean curvature (Bruno Bianchini, Giulio Colombo, Marco Magliaro et al., 2020)7Bianchini 2020 Characteristics of conformal Ricci soliton on warped product spaces (Dipen Ganguly, Nirabhra Basu and Arindam Bhattacharyya, 2021)5Ganguly 2021 Gradient Ricci solitons carrying a closed conformal vector field (J. F. Siva Filho and R. Sharma, 2021)6Filho 2021 On mean curvature flow solitons in the sphere (Marco Magliaro, Luciano Mari, Fernanda Roing et al., 2025)8Magliaro 2025 Translating solitons to higher order mean curvature flows in Riemannian products (Jorge Herbert Soares de Lira and Rafael Rocha de Farias, 2026)2de Lira 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Remarks on mean curvature flow solitons in warped products

    Giulio Colombo, Luciano Mari and Marco RigoliWider flows

    Topics Rotational symmetry & warped products · Same author Marco Rigoli

  2. 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 · Same author Jorge H. de Lira · Shared terms “Riemannian product”

  3. Translating solitons in Riemannian products

    Jorge H. de Lira and Francisco MartinWider flows

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

  4. Long Time Existence of IMCF on Metrics Conformal to Warped Product Manifolds

    Brian AllenWider flows

    Topics Rotational symmetry & warped products · Shared terms “conformal vector”, “vector field”

  5. Characteristics of conformal Ricci soliton on warped product spaces

    Dipen Ganguly, Nirabhra Basu and Arindam Bhattacharyya

    Topics Rotational symmetry & warped products · Shared terms “product spaces”, “conformal vector”, “vector field”

  6. Gradient Ricci solitons carrying a closed conformal vector field

    J. F. Siva Filho and R. Sharma

    Topics Rotational symmetry & warped products · Shared terms “conformal vector”, “vector field”

  7. Recent rigidity results for graphs with prescribed mean curvature

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

    Topics Rotational symmetry & warped products · Same author Marco Rigoli

  8. On mean curvature flow solitons in the sphere

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

    Topics Rotational symmetry & warped products · Shared terms “vector field”

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) 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.