Paper · Nov 15, 2025

Quasi-Einstein Metrics and a curvature identity associated with the Ricci flow

Atreyee Bhattacharya, Sayoojya Prakash

math.DGarXiv:2511.12144v2
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Abstract

Quasi-Einstein manifolds are well-studied generalizations of Einstein manifolds. This includes gradient Ricci solitons and has a natural correspondence with the warped product Einstein manifolds. A quasi-Einstein metric is said to be rigid when it reduces to an Einstein metric. On a different note, Einstein metrics can be viewed as fixed points of the Ricci flow up to homothety. While gradient Ricci solitons are generalized fixed points of the Ricci flow, not much is known, in general, about the evolution of quasi-Einstein metrics under the Ricci flow. In this paper, we employ an identity associated to the evolution of curvature along the Ricci flow, to conclude the rigidity of certain closed quasi-Einstein manifolds.

Comments: Updated version

Ricci flowRicci solitonsEinstein metricsAlmost & η-Ricci solitonsStability & uniquenessRotational symmetry & warped products

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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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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ricci solitons as critical points of quadratic curvature functionals

    Atreyee Bhattacharya and Sayoojya Prakash PDF

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

  2. A Characterization of Quasi-Einstein Metrics

    Antonio Airton Freitas Filho PDF

    Topics Almost & η-Ricci solitons, Rotational symmetry & warped products, Einstein metrics · Shared terms “quasi Einstein”

  3. The extended quasi-Einstein manifolds with generalised Ricci solitons

    Zhiming Huang, Weijun Lu and Fuhong Su PDF

    Topics Almost & η-Ricci solitons, Einstein metrics, Ricci solitons · Shared terms “quasi Einstein”, “generalization”

  4. Direct proofs for spaces with cyclic parallel Ricci tensor

    Fengyuan Zhang PDF

    Topics Almost & η-Ricci solitons, Einstein metrics, Stability & uniqueness · Shared terms “quasi Einstein”, “identity”, “gradient Ricci”

  5. Diameter estimates and Hitchin-Thorpe inequality for four-dimensional compact Quasi-Einstein manifolds

    Samuel Belo PDF

    Topics Almost & η-Ricci solitons, Einstein metrics, Ricci solitons · Shared terms “quasi Einstein”

  6. The rigity of spaces with cyclic parallel Ricci tensor

    Fengyuan Zhang PDF

    Topics Almost & η-Ricci solitons, Einstein metrics, Ricci solitons · Shared terms “quasi Einstein”, “identity”

  7. On noncompact warped product Ricci solitons

    Valter Borges PDF

    Topics Rotational symmetry & warped products, Einstein metrics, Stability & uniqueness · Shared terms “gradient Ricci”

  8. Almost Ricci solitons on Finsler spaces

    Qiaoling Xia PDF

    Topics Almost & η-Ricci solitons, Einstein metrics, Stability & uniqueness · Shared terms “gradient 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 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) 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) Almost & η-Ricci solitons Variations on the Ricci soliton equation: let the constant λ become a function (almost solitons), add a term along a contact form (η-Ricci… Start with: Ricci almost solitons (Pigola et al., 2010)

In the guidebook

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

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

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

  3. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.