Paper · Jun 11, 2026

Ricci solitons as critical points of quadratic curvature functionals

Atreyee Bhattacharya, Sayoojya Prakash

math.DGarXiv:2606.13066
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Abstract

Rigidity, stability and local minimizing properties of Einstein metrics as critical points of quadratic Riemannian functionals defined by -norms of Ricci curvature, scalar curvature, Weyl curvature and Riemannian curvature have been extensively studied. However, there are non-Einstein critical points of these functionals that are not so well understood. In this paper, we study Ricci solitons, a generalization of Einstein metrics, that are critical points of a special quadratic curvature functional and analyze their rigidity.

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Submitted
Jun 11, 2026
Subjects
math.DG

Comments: 8 pages

Ricci solitonsEinstein metricsScalar curvatureStability & uniqueness

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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 Geometric Flows of G₂-Structures on 3-Sasakian 7-Manifolds (Aaron Kennon and Jason D. Lotay, 2022)3Kennon 2022 Linear stability of compact shrinking Ricci solitons (Huai-Dong Cao and Meng Zhu, 2023)4Cao 2023 A rigidity result for axisymmetric toric Ricci solitons (Shiqiao Zhang, 2024)7Zhang 2024 Ricci flow and the scalar curvature rigidity of Einstein manifolds (Klaus Kroencke, 2025)5Kroencke 2025 Quasi-Einstein Metrics and a curvature identity associated with the Ricci flow (Atreyee Bhattacharya and Sayoojya Prakash, 2025)1Bhattacharya 2025 On the Stability of Einstein Manifolds with Boundary (Rasmus Jouttijärvi, 2026)8Jouttijärvi 2026 Kähler–Ricci Shrinkers with Constant Scalar Curvature Are Rigid (Wenqi Li, Jianyu Ou, Guoqiang Wu et al., 2026)6Li 2026 Rigidity of Constant-Scalar-Curvature Ricci Shrinkers with Totally Geodesic Focal Set (Fengjiang Li, Wenqi Li, Jianyu Ou et al., 2026)2Li 2026
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  1. Quasi-Einstein Metrics and a curvature identity associated with the Ricci flow

    Atreyee Bhattacharya and Sayoojya Prakash

    Topics Einstein metrics, Stability & uniqueness, Ricci solitons · Same authors Atreyee Bhattacharya, Sayoojya Prakash · Shared terms “generalization”

  2. Rigidity of Constant-Scalar-Curvature Ricci Shrinkers with Totally Geodesic Focal Set

    Fengjiang Li, Wenqi Li, Jianyu Ou et al.

    Topics Einstein metrics, Scalar curvature, Stability & uniqueness

  3. Geometric Flows of -Structures on 3-Sasakian 7-Manifolds

    Aaron Kennon and Jason D. Lotay

    Topics Einstein metrics, Stability & uniqueness · Shared terms “critical points”

  4. Linear stability of compact shrinking Ricci solitons

    Huai-Dong Cao and Meng Zhu

    Topics Einstein metrics, Stability & uniqueness, Ricci solitons

  5. Ricci flow and the scalar curvature rigidity of Einstein manifolds

    Klaus Kroencke

    Topics Einstein metrics, Scalar curvature, Stability & uniqueness

  6. Kähler–Ricci Shrinkers with Constant Scalar Curvature Are Rigid

    Wenqi Li, Jianyu Ou, Guoqiang Wu et al.

    Topics Einstein metrics, Scalar curvature, Stability & uniqueness

  7. A rigidity result for axisymmetric toric Ricci solitons

    Shiqiao Zhang

    Topics Einstein metrics, Stability & uniqueness, Ricci solitons

  8. On the Stability of Einstein Manifolds with Boundary

    Rasmus Jouttijärvi

    Topics Einstein metrics, Scalar curvature, Stability & uniqueness

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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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) 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)

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.