Paper · May 2, 2022

Closed self-similar solutions to flows by negative powers of curvature

Shanze Gao

math.DGWider flowsarXiv:2205.00927v3
View PDF View HTML arXiv page

As on arXiv

Abstract

In some warped product manifolds including space forms, we consider closed self-similar solutions to curvature flows whose speeds are negative powers of mean curvature, Gauss curvature and other curvature functions with suitable properties. We prove such self-similar solutions, not necessarily strictly convex for some cases, must be slices of warped product manifolds. A new auxiliary function is the key of the proofs.

View PDF View HTML arXiv page
Submitted
May 2, 2022
Last revised
Sep 3, 2023 (v3)
Subjects
math.DG

Comments: 15 pages, to appear in J. Geom. Anal

Other curvature flowsSphere theoremsRotational 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 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 Non-homothetic convex ancient solutions for flows by high powers of curvature (Susanna Risa and Carlo Sinestrari, 2022)3Risa 2022 Mean curvature flows of graphs sliding off to infinity in warped product manifolds (Naotoshi Fujihara, 2023)2Fujihara 2023 Uniqueness and Symmetry of Self-Similar Solutions of Curvature Flows in Warped Product Spaces (Frederick Tsz-Ho Fong, 2024)1Fong 2024 Isotropy invariant graphical mean curvature flows in warped products (Naotoshi Fujihara and Naoyuki Koike, 2024)6Fujihara 2024 On the geometry of Riemannian warped product maps (Jyoti Yadav, Harmandeep Kaur and Gauree Shanker, 2025)5Yadav 2025 Geometric properties of second Ricci solitons (Masoumeh Khalili, Ghodratallah Fasihi-Ramandi and Shahroud Azami, 2025)4Khalili 2025 Translating solitons to higher order mean curvature flows in Riemannian products (Jorge Herbert Soares de Lira and Rafael Rocha de Farias, 2026)8de Lira 2026 Singular Rotational Self-Similar Tori for Odd σₖ-Curvature Flows (Haoxuan Cheng, Junqi Lai and Guoxin Wei, 2026)7Cheng 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Uniqueness and Symmetry of Self-Similar Solutions of Curvature Flows in Warped Product Spaces

    Frederick Tsz-Ho FongWider flows

    Topics Rotational symmetry & warped products, Other curvature flows

  2. Mean curvature flows of graphs sliding off to infinity in warped product manifolds

    Naotoshi FujiharaWider flows

    Topics Rotational symmetry & warped products

  3. Non-homothetic convex ancient solutions for flows by high powers of curvature

    Susanna Risa and Carlo SinestrariWider flows

    Topics Rotational symmetry & warped products, Other curvature flows · Shared terms “auxiliary”

  4. Geometric properties of second Ricci solitons

    Masoumeh Khalili, Ghodratallah Fasihi-Ramandi and Shahroud Azami

    Topics Rotational symmetry & warped products

  5. On the geometry of Riemannian warped product maps

    Jyoti Yadav, Harmandeep Kaur and Gauree Shanker

    Topics Rotational symmetry & warped products

  6. Isotropy invariant graphical mean curvature flows in warped products

    Naotoshi Fujihara and Naoyuki KoikeWider flows

    Topics Rotational symmetry & warped products

  7. Singular Rotational Self-Similar Tori for Odd -Curvature Flows

    Haoxuan Cheng, Junqi Lai and Guoxin WeiWider flows

    Topics Rotational symmetry & warped products, Other curvature flows

  8. 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

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.

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) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988) 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. 11A.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.