Paper · Aug 22, 2023

Classification of ruled surfaces as homothetic self-similar solutions of the inverse mean curvature flow in the Lorentz-Minkowski 3-space

Gregório Silva Neto, Vanessa Silva

math.DGWider flowsarXiv:2308.11679
PDF Abstract page

As on arXiv

Abstract

In this paper, we classify the nondegenerate ruled surfaces in the three-dimensional Lorentz-Minkowski space that are homothetic self-similar solutions for the inverse mean curvature flow. This classification shows the existence of two classes of non-cylindrical homothetic solitons: one with lightlike rulings and another one with non-lightlike rulings.

Comments: 26 pages, 6 figures. Final version, to appear in the Bulletin of the Brazilian Mathematical Society, New Series. arXiv admin note: substantial text overlap with arXiv:2308.11457

Mean curvature flowOther curvature flowsInverse mean curvature flow3-manifolds

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 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 The streamlines of ∞-harmonic functions obey the inverse mean curvature flow (Roger Moser, 2021)6Moser 2021 An anisotropic inverse mean curvature flow for spacelike graphic curves in Lorentz-Minkowski plane ℝ²₁ (Ya Gao, Chenyang Liu and Jing Mao, 2021)2Gao 2021 Graphical translating solitons for the inverse mean curvature flow and isoparametric functions (Tomoki Fujii, 2022)7Fujii 2022 Inverse mean curvature flow and Ricci-pinched three-manifolds (Gerhard Huisken and Thomas Koerber, 2023)3Huisken 2023 A topological gap theorem for the π₂-systole of positive scalar curvature 3-manifolds (Kai Xu, 2023)5Xu 2023 Ruled surfaces as translating solitons of the inverse mean curvature flow in the three-dimensional Lorentz-Minkowski space (Gregório Silva Neto and Vanessa Silva, 2023)1Neto 2023 Scalar curvature bounds for 3D continuous metrics through the Inverse Mean Curvature Flow (Mattia Fogagnolo, Giorgio Gatti and Alessandra Pluda, 2026)4Fogagnolo 2026 A Positive Mass Type Theorem for a Singular Toroidal Static Model (Zhixin Wang, 2026)8Wang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ruled surfaces as translating solitons of the inverse mean curvature flow in the three-dimensional Lorentz-Minkowski space

    Gregório Silva Neto and Vanessa SilvaWider flows PDF

    Topics Inverse mean curvature flow, 3-manifolds, Other curvature flows · Same authors Gregório Silva Neto, Vanessa Silva · Shared terms “ruled surfaces”, “Lorentz Minkowski”, “dimensional Lorentz”

  2. An anisotropic inverse mean curvature flow for spacelike graphic curves in Lorentz-Minkowski plane

    Ya Gao, Chenyang Liu and Jing MaoWider flows PDF

    Topics Inverse mean curvature flow, Other curvature flows · Shared terms “Lorentz Minkowski”, “dimensional Lorentz”

  3. Inverse mean curvature flow and Ricci-pinched three-manifolds

    Gerhard Huisken and Thomas Koerber PDF

    Topics Inverse mean curvature flow, 3-manifolds, Other curvature flows

  4. Scalar curvature bounds for 3D continuous metrics through the Inverse Mean Curvature Flow

    Mattia Fogagnolo, Giorgio Gatti and Alessandra PludaWider flows PDF

    Topics Inverse mean curvature flow, 3-manifolds, Other curvature flows

  5. A topological gap theorem for the -systole of positive scalar curvature 3-manifolds

    Kai XuWider flows PDF

    Topics Inverse mean curvature flow, 3-manifolds, Other curvature flows

  6. The streamlines of -harmonic functions obey the inverse mean curvature flow

    Roger MoserWider flows PDF

    Topics Inverse mean curvature flow, Other curvature flows

  7. Graphical translating solitons for the inverse mean curvature flow and isoparametric functions

    Tomoki FujiiWider flows PDF

    Topics Inverse mean curvature flow, Other curvature flows

  8. A Positive Mass Type Theorem for a Singular Toroidal Static Model

    Zhixin WangWider flows PDF

    Topics Inverse mean curvature flow, 3-manifolds, Other curvature flows

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. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 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) Inverse mean curvature flow Wider flows A hypersurface expands with normal speed 1/H, the reciprocal of its mean curvature. Huisken and Ilmanen's weak version keeps the Hawking mass… Start with: Flow of nonconvex hypersurfaces into spheres (Gerhardt, 1990) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 10A.1
    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

  2. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  3. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.