Paper · Oct 7, 2020

Nonhomogeneous expanding flows in hyperbolic spaces

Giuseppe Pipoli

math.DGWider flowsarXiv:2010.03308
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Abstract

A recent paper [CGT] studies the evolution of star-shaped mean convex hypersurfaces of the Euclidean space by a class of nonhomogeneous expanding curvature flows. In the present paper we consider the same problem in the real, complex and quaternionic hyperbolic spaces, investigating how the richer geometry of the ambient space affects the evolution. In every case the initial conditions are preserved and the long time existence of the flow is proven. The geometry of the ambient space influences the asymptotic behaviour of the flow: after a suitable rescaling the induced metric converges to a conformal multiple of the standard Riemannian round metric of the sphere if the ambient manifold is the real hyperbolic space, otherwise it converges to a conformal multiple of the standard sub-Riemannian metric on the odd-dimensional sphere. Finally, in every cases, we are able to construct infinitely many examples such that the limit does not have constant scalar curvature.

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Submitted
Oct 7, 2020
Subjects
math.DG

Comments: 25 pages, comments are welcome

Other curvature flowsScalar curvatureHomogeneous spacesLong-time existence & convergence

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 Inverse mean curvature flow in complex hyperbolic space (Giuseppe Pipoli, 2016)2Pipoli 2016 Volume preserving non homogeneous mean curvature flow in hyperbolic space (Maria Chiara Bertini and Giuseppe Pipoli, 2016)3Bertini 2016 Inverse mean curvature flow in quaternionic hyperbolic space (Giuseppe Pipoli, 2017)1Pipoli 2017 Invariant translators of the Heisenberg group (Giuseppe Pipoli, 2018)7Pipoli 2018 Invariant translators of the Solvable group (Giuseppe Pipoli, 2019)8Pipoli 2019 Contracting self-similar solutions of nonhomogeneous curvature flows (James McCoy, 2020)6McCoy 2020 Volume preserving nonhomogeneous Gauss curvature flow in hyperbolic space (Yong Wei, Bo Yang and Tailong Zhou, 2024)4Wei 2024 Long time behavior of a class of non-homogeneous anisotropic fully nonlinear curvature flows (Weimin Sheng and Jiazhuo Yang, 2025)5Sheng 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Inverse mean curvature flow in quaternionic hyperbolic space

    Giuseppe PipoliWider flows

    Topics Scalar curvature, Other curvature flows · Same author Giuseppe Pipoli · Shared terms “quaternionic hyperbolic”, “hyperbolic space”, “conformal multiple”

  2. Inverse mean curvature flow in complex hyperbolic space

    Giuseppe PipoliWider flows

    Topics Other curvature flows · Same author Giuseppe Pipoli · Shared terms “hyperbolic space”, “conformal multiple”, “standard sub”

  3. Volume preserving non homogeneous mean curvature flow in hyperbolic space

    Maria Chiara Bertini and Giuseppe PipoliWider flows

    Topics Homogeneous spaces, Long-time existence & convergence, Other curvature flows · Same author Giuseppe Pipoli · Shared terms “hyperbolic space”

  4. Volume preserving nonhomogeneous Gauss curvature flow in hyperbolic space

    Yong Wei, Bo Yang and Tailong ZhouWider flows

    Topics Homogeneous spaces, Long-time existence & convergence, Other curvature flows · Shared terms “nonhomogeneous”, “hyperbolic space”, “convex hypersurfaces”

  5. Long time behavior of a class of non-homogeneous anisotropic fully nonlinear curvature flows

    Weimin Sheng and Jiazhuo YangWider flows

    Topics Homogeneous spaces, Long-time existence & convergence, Other curvature flows · Shared terms “star shaped”, “convex hypersurfaces”

  6. Contracting self-similar solutions of nonhomogeneous curvature flows

    James McCoyWider flows

    Topics Homogeneous spaces, Other curvature flows · Shared terms “nonhomogeneous”, “convex hypersurfaces”

  7. Invariant translators of the Heisenberg group

    Giuseppe PipoliWider flows

    Same author Giuseppe Pipoli · Shared terms “standard Riemannian”, “standard sub”, “metric converges”

  8. Invariant translators of the Solvable group

    Giuseppe PipoliWider flows

    Topics Homogeneous spaces · Same author Giuseppe Pipoli · Shared terms “ambient space”, “Euclidean space”

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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.

  2. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  3. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.