Paper · Jul 24, 2026

Non-compact Ricci Solitons of Finite Volume with Potential Field of Constant Length

Hemangi Madhusudan Shah, Sharief Deshmukh, Mohammad Aqib

math.DGarXiv:2607.22394v2
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Abstract

We study non-compact Ricci solitons of finite volume whose potential vector field has constant length. Under the assumptions that the scalar curvature is constant along the integral curves of the potential field and that a natural divergence term is integrable on the unit tangent bundle, we prove that such Ricci solitons are necessarily trivial. As applications, we characterise -dimensional Ricci solitons of constant length and of finite volume on almost -cosymplectic manifolds. In fact, we show that these spaces are indeed compact quotients of Euclidean spaces. We also study complete, non-compact and finite volume contact metric manifolds in dimension admitting a Reeb vector field as a Ricci soliton. We show that it is a compact spherical space form. We include some examples to illustrate the necessity of the finite-volume and constant-length assumptions.

Ricci solitonsSphere theoremsScalar curvatureBoundaries & non-compactSasakian & contact geometry

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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 Kenmotsu metric as conformal η-Ricci soliton (Dipen Ganguly, 2021)6Ganguly 2021 Gradient pseudo-Ricci solitons of real hypersurfaces (Mayuko Kon, 2021)7Kon 2021 Nature of Some Solitons on Almost coKähler Manifolds and Asymptotically Harmonic Manifolds (Paritosh Ghosh, Hemangi Madhusudan Shah and Arindam Bhattacharyya, 2023)5Ghosh 2023 η-Ricci solitons and η-Einstein metrics on weak β-Kenmotsu f-manifolds (Vladimir Rovenski, 2024)4Rovenski 2024 Minimal surface entropy and applications of Ricci flow on finite-volume hyperbolic 3-manifolds (Ruojing Jiang and Franco Vargas Pallete, 2025)8Jiang 2025 Characterizations of Almost Ricci Bourguignon Solitons (Mohammad Aqib, Hemangi Madhusudan Shah and Dhriti Sundar Patra, 2025)1Aqib 2025 Ricci solitons from the perspectives of energy function (Shubham Yadav and Hemangi Madhusudan Shah, 2026)3Yadav 2026 η- Hyperbolic Ricci Solitons on the Unit Tangent Bundle of the Hyperbolic Strip (Amadou Sy, Ameth Ndiaye and Ghodratallah Fasihi-Ramandi, 2026)2Sy 2026
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  1. Characterizations of Almost Ricci Bourguignon Solitons

    Mohammad Aqib, Hemangi Madhusudan Shah and Dhriti Sundar Patra PDF

    Topics Ricci solitons · Same authors Hemangi Madhusudan Shah, Mohammad Aqib

  2. Hyperbolic Ricci Solitons on the Unit Tangent Bundle of the Hyperbolic Strip

    Amadou Sy, Ameth Ndiaye and Ghodratallah Fasihi-Ramandi PDF

    Topics Sasakian & contact geometry, Ricci solitons · Shared terms “unit tangent”, “tangent bundle”, “vector field”

  3. Ricci solitons from the perspectives of energy function

    Shubham Yadav and Hemangi Madhusudan Shah PDF

    Topics Scalar curvature, Ricci solitons · Same author Hemangi Madhusudan Shah · Shared terms “vector field”

  4. -Ricci solitons and -Einstein metrics on weak -Kenmotsu -manifolds

    Vladimir Rovenski PDF

    Topics Sasakian & contact geometry, Scalar curvature, Ricci solitons · Shared terms “Reeb vector”, “vector field”, “potential vector”

  5. Nature of Some Solitons on Almost coKähler Manifolds and Asymptotically Harmonic Manifolds

    Paritosh Ghosh, Hemangi Madhusudan Shah and Arindam Bhattacharyya PDF

    Topics Scalar curvature, Ricci solitons · Same author Hemangi Madhusudan Shah · Shared terms “admitting”

  6. Kenmotsu metric as conformal -Ricci soliton

    Dipen Ganguly PDF

    Topics Sasakian & contact geometry, Ricci solitons · Shared terms “Reeb vector”, “vector field”, “potential vector”

  7. Gradient pseudo-Ricci solitons of real hypersurfaces

    Mayuko Kon PDF

    Topics Sphere theorems, Ricci solitons · Shared terms “vector field”, “space forms”

  8. Minimal surface entropy and applications of Ricci flow on finite-volume hyperbolic 3-manifolds

    Ruojing Jiang and Franco Vargas Pallete PDF

    Topics Sphere theorems, Scalar curvature · Shared terms “finite volume”

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.

Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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) 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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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. 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.