Paper · Apr 28, 2026

Contraction of hypersurfaces with positive sectional curvature in hyperbolic space

Tianci Luo, Yong Wei, Rong Zhou

math.DGWider flowsarXiv:2604.25513
PDF Abstract page

As on arXiv

Abstract

We study contracting curvature flows of compact hypersurfaces with positive sectional curvature in hyperbolic space . The speed is assumed to be homogeneous of degree one in the principal curvatures and to satisfy certain conditions. This class of flows includes the th mean curvature flow as a special case. We show that if the initial hypersurface has positive sectional curvature, then this property is preserved along the flow, and the evolving hypersurface contracts to a round point in finite time.

Comments: 25 pages

Mean curvature flowOther curvature flowsSingularitiesSphere theoremsHomogeneous spaces

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 Ricci flow does not preserve positive sectional curvature in dimension four (Renato G. Bettiol and Anusha M. Krishnan, 2021)6Bettiol 2021 Positive sectional curvature is not preserved under the Ricci flow in dimensions seven and thirteen (David González-Álvaro and Masoumeh Zarei, 2024)3González-Álv 2024 Volume preserving nonhomogeneous Gauss curvature flow in hyperbolic space (Yong Wei, Bo Yang and Tailong Zhou, 2024)8Wei 2024 Ricci Flow Preserves Positive Sectional Curvature on Homogeneous Spheres (Jason DeVito, David González-Álvaro and Masoumeh Zarei, 2026)4DeVito 2026 Free boundary flows by powers of the Gauss curvature in the unit ball (Tianci Luo, Yong Wei and Rong Zhou, 2026)5Luo 2026 Weinstock inequalities for outward-minimizing domains (Chaoqun Gao, Yong Wei and Rong Zhou, 2026)7Gao 2026 Positive τ-bi-Ricci curvature and Mean curvature flow with surgery in hyperbolic space (Tianci Luo, Yong Wei and Rong Zhou, 2026)2Luo 2026 Curvature quotient flows and Alexandrov–Fenchel inequalities in hyperbolic space (Tianci Luo, Yong Wei, Ruijia Zhang et al., 2026)1Luo 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Curvature quotient flows and Alexandrov–Fenchel inequalities in hyperbolic space

    Tianci Luo, Yong Wei, Ruijia Zhang et al.Wider flows PDF

    Topics Sphere theorems, Other curvature flows · Same authors Tianci Luo, Yong Wei, Rong Zhou · Shared terms “positive sectional”, “sectional curvature”, “hyperbolic space”

  2. Positive -bi-Ricci curvature and Mean curvature flow with surgery in hyperbolic space

    Tianci Luo, Yong Wei and Rong ZhouWider flows PDF

    Topics Mean curvature flow · Same authors Tianci Luo, Yong Wei, Rong Zhou · Shared terms “hyperbolic space”, “round point”, “initial hypersurface”

  3. Positive sectional curvature is not preserved under the Ricci flow in dimensions seven and thirteen

    David González-Álvaro and Masoumeh Zarei PDF

    Topics Sphere theorems · Shared terms “positive sectional”, “sectional curvature”, “preserved”

  4. Ricci Flow Preserves Positive Sectional Curvature on Homogeneous Spheres

    Jason DeVito, David González-Álvaro and Masoumeh Zarei PDF

    Topics Sphere theorems, Homogeneous spaces · Shared terms “positive sectional”, “sectional curvature”

  5. Free boundary flows by powers of the Gauss curvature in the unit ball

    Tianci Luo, Yong Wei and Rong ZhouWider flows PDF

    Topics Other curvature flows, Singularities · Same authors Tianci Luo, Yong Wei, Rong Zhou · Shared terms “contracts”, “finite time”

  6. Ricci flow does not preserve positive sectional curvature in dimension four

    Renato G. Bettiol and Anusha M. Krishnan PDF

    Topics Sphere theorems, Homogeneous spaces · Shared terms “positive sectional”, “sectional curvature”

  7. Weinstock inequalities for outward-minimizing domains

    Chaoqun Gao, Yong Wei and Rong ZhouWider flows PDF

    Topics Other curvature flows · Same authors Yong Wei, Rong Zhou · Shared terms “hyperbolic space”

  8. Volume preserving nonhomogeneous Gauss curvature flow in hyperbolic space

    Yong Wei, Bo Yang and Tailong ZhouWider flows PDF

    Topics Homogeneous spaces, Other curvature flows · Same author Yong Wei · Shared terms “hyperbolic 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.

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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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)

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.

  2. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

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

  3. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.