Paper · Aug 29, 2025

On the stability of Ricci flow on hyperbolic 3-manifolds of finite volume

Ruojing Jiang, Franco Vargas Pallete

math.DGarXiv:2509.00188
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Abstract

On a hyperbolic 3-manifold of finite volume, we prove that if the initial metric is sufficiently close to the hyperbolic metric , then the normalized Ricci-DeTurck flow exists for all time and converges exponentially fast to in a weighted Hölder norm. A key ingredient of our approach is the application of interpolation theory. Furthermore, this result is a valuable tool for investigating minimal surface entropy, which quantifies the growth rate of the number of closed minimal surfaces in terms of genus. We explore this in [17].

Ricci flow3-manifoldsStability & uniquenessLong-time existence & convergenceHarmonic maps & DeTurck trickMinimal & CMC surfaces

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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 Minimal Surface Entropy and Average Area Ratio (Ben Lowe and Andre Neves, 2021)3Lowe 2021 Convergence stability for Ricci flow on manifolds with bounded geometry (Eric Bahuaud, Christine Guenther, James Isenberg et al., 2023)2Bahuaud 2023 Ricci-DeTurck Flow from Initial Metric with Morrey-type Integrability Condition (Man-Chun Lee and Stephen Shang Yi Liu, 2024)8Lee 2024 Stability of the area preserving mean curvature flow in asymptotic Schwarzschild space (Yaoting Gui, Yuqiao Li and Jun Sun, 2024)6Gui 2024 On the long-time limit of the mean curvature flow in closed manifolds (Alexander Mramor and Ao Sun, 2024)5Mramor 2024 An alternative solvability criterion for the Dirichlet problem for the minimal surface equation and an application to the mean curvature flow (Ari J. Aiolfi, Giovanni da Silva Nunes, Jaime Ripoll et al., 2025)4Aiolfi 2025 Minimal surface entropy and applications of Ricci flow on finite-volume hyperbolic 3-manifolds (Ruojing Jiang and Franco Vargas Pallete, 2025)1Jiang 2025 Integrable Deformations and Stability of the Ricci Flow (Maxwell Stolarski and Alex Waldron, 2026)7Stolarski 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Minimal surface entropy and applications of Ricci flow on finite-volume hyperbolic 3-manifolds

    Ruojing Jiang and Franco Vargas Pallete PDF

    Topics Minimal & CMC surfaces, 3-manifolds · Same authors Ruojing Jiang, Franco Vargas Pallete · Shared terms “finite volume”, “surface entropy”, “hyperbolic metric”

  2. Convergence stability for Ricci flow on manifolds with bounded geometry

    Eric Bahuaud, Christine Guenther, James Isenberg et al. PDF

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “hyperbolic metric”, “Hölder norm”, “sufficiently close”

  3. Minimal Surface Entropy and Average Area Ratio

    Ben Lowe and Andre Neves PDF

    Topics Minimal & CMC surfaces, 3-manifolds · Shared terms “surface entropy”, “hyperbolic metric”

  4. An alternative solvability criterion for the Dirichlet problem for the minimal surface equation and an application to the mean curvature flow

    Ari J. Aiolfi, Giovanni da Silva Nunes, Jaime Ripoll et al.Wider flows PDF

    Topics Minimal & CMC surfaces, Harmonic maps & DeTurck trick

  5. On the long-time limit of the mean curvature flow in closed manifolds

    Alexander Mramor and Ao SunWider flows PDF

    Topics Minimal & CMC surfaces, 3-manifolds

  6. Stability of the area preserving mean curvature flow in asymptotic Schwarzschild space

    Yaoting Gui, Yuqiao Li and Jun SunWider flows PDF

    Topics Minimal & CMC surfaces, 3-manifolds, Long-time existence & convergence · Shared terms “exponentially fast”, “converges exponentially”, “sufficiently”

  7. Integrable Deformations and Stability of the Ricci Flow

    Maxwell Stolarski and Alex Waldron PDF

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “weighted Hölder”

  8. Ricci-DeTurck Flow from Initial Metric with Morrey-type Integrability Condition

    Man-Chun Lee and Stephen Shang Yi Liu PDF

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “initial metric”

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) 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. 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.