Paper · Jun 12, 2026

Ricci solitons from the perspectives of energy function

Shubham Yadav, Hemangi Madhusudan Shah

math.DGarXiv:2606.14336v2
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Abstract

This article explores to what extent the geometry of gradient Ricci solitons extends to non-gradient Ricci solitons. The primary tool is the energy function of the soliton. We study consequences of various bounds on . Under mild assumptions on the scalar curvature, we prove a weighted -Liouville type theorem for both the usual Laplacian and the drifted Laplacian associated to soliton vector field , the former of which implies that Ricci solitons with bounded energy function have at most one nonparabolic end. Finally, we show that the measure is finite for complete shrinking Ricci solitons, partially generalizing a result of Aaron Naber. As a consequence, non-gradient shrinking Ricci solitons also have finite fundamental groups, as in the gradient case.

Comments: 21 pages; The manuscript is modified

Ricci solitonsScalar curvature

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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.

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  1. Three circles theorems and Liouville type theorems

    Run-Qiang Jian and Zhu-Hong Zhang PDF

    Topics Scalar curvature, Ricci solitons · Shared terms “Liouville type”, “type theorem”, “gradient shrinking”

  2. Liouville Theorem on Ricci shrinkers with constant scalar curvature and its application

    Weixiong Mai and Jianyu Ou PDF

    Topics Scalar curvature, Ricci solitons · Shared terms “shrinking Ricci”, “Liouville”, “gradient shrinking”

  3. 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

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

    Hemangi Madhusudan Shah, Sharief Deshmukh and Mohammad Aqib PDF

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

  5. Some rigidity results on shrinking gradient Ricci soliton

    Jianyu Ou, Yuanyuan Qu and Guoqiang Wu PDF

    Topics Scalar curvature, Ricci solitons · Shared terms “gradient Ricci”, “complete shrinking”, “generalizing”

  6. Rigidity of shrinking gradient ricci soliton with constant scalar curvature

    Fengjiang Li, Yuanyuan Qu and Guoqiang Wu PDF

    Topics Scalar curvature, Ricci solitons · Shared terms “gradient Ricci”, “complete shrinking”

  7. Rigidity of Five-dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

    Fengjiang Li, Jianyu Ou, Yuanyuan Qu et al. PDF

    Topics Scalar curvature, Ricci solitons · Shared terms “gradient Ricci”, “shrinking Ricci”, “gradient shrinking”

  8. Rigidity results for shrinking and expanding Ricci solitons

    Benedito Leandro and Jeferson Poveda PDF

    Topics Scalar curvature, Ricci solitons · Shared terms “shrinking Ricci”

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) 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)

In the guidebook

  1. 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.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

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

  3. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.