Paper · Sep 29, 2025

A Characterization of Quasi-Einstein Metrics

Antonio Airton Freitas Filho

math.DGarXiv:2509.25132v2
PDF Abstract page

As on arXiv

Abstract

We study the modified Ricci solitons as a new class of Einstein type metrics that contains both Ricci solitons and -quasi-Einstein metrics. This class is closely related to the construction of the Ricci solitons that are realised as warped products. A modified Ricci soliton appears as part of a special solution of the modified Ricci-harmonic flow, which result a new characterization of -quasi-Einstein metrics. We also study the modified Ricci almost solitons. In the spirit of the Lichnerowicz and Obata first eigenvalue theorems, we prove that in the class of compact Riemannian manifolds with constant scalar curvature the standard sphere with a structure of gradient modified Ricci almost soliton is rigid under some specific geometric conditions. Moreover, we display an example of modified Ricci-harmonic soliton.

Ricci flowRicci solitonsEinstein metricsAlmost & η-Ricci solitonsScalar curvatureRotational symmetry & warped productsEigenvalues & spectrum

Its neighbourhood

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 Classification of generalized Yamabe solitons under vanishing conditions on the Weyl, Cotton, and Cao-Chen tensors (Shun Maeta, 2021)7Maeta 2021 On noncompact warped product Ricci solitons (Valter Borges, 2022)4Borges 2022 The extended quasi-Einstein manifolds with generalised Ricci solitons (Zhiming Huang, Weijun Lu and Fuhong Su, 2022)3Huang 2022 Eigenvalue lower bounds and splitting for modified Ricci flow (Tobias Holck Colding and William P. Minicozzi, 2023)2Colding 2023 Nature of Some Solitons on Almost coKähler Manifolds and Asymptotically Harmonic Manifolds (Paritosh Ghosh, Hemangi Madhusudan Shah and Arindam Bhattacharyya, 2023)8Ghosh 2023 Almost Ricci solitons on Finsler spaces (Qiaoling Xia, 2024)6Xia 2024 Local structure of gradient almost Ricci solitons with harmonic Weyl tensor (Valter Borges, Matheus Andrade Ribeiro de Moura Horácio and João Paulo dos Santos, 2025)5Borges 2025 Quasi-Einstein Metrics and a curvature identity associated with the Ricci flow (Atreyee Bhattacharya and Sayoojya Prakash, 2025)1Bhattacharya 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Quasi-Einstein Metrics and a curvature identity associated with the Ricci flow

    Atreyee Bhattacharya and Sayoojya Prakash PDF

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

  2. Eigenvalue lower bounds and splitting for modified Ricci flow

    Tobias Holck Colding and William P. Minicozzi PDF

    Topics Eigenvalues & spectrum · Shared terms “modified Ricci”

  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”

  4. On noncompact warped product Ricci solitons

    Valter Borges PDF

    Topics Eigenvalues & spectrum, Rotational symmetry & warped products, Einstein metrics

  5. Local structure of gradient almost Ricci solitons with harmonic Weyl tensor

    Valter Borges, Matheus Andrade Ribeiro de Moura Horácio and João Paulo dos Santos PDF

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

  6. Almost Ricci solitons on Finsler spaces

    Qiaoling Xia PDF

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

  7. Classification of generalized Yamabe solitons under vanishing conditions on the Weyl, Cotton, and Cao-Chen tensors

    Shun Maeta PDF

    Topics Almost & η-Ricci solitons, Rotational symmetry & warped products, Scalar curvature · Shared terms “Einstein type”

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

    Paritosh Ghosh, Hemangi Madhusudan Shah and Arindam Bhattacharyya PDF

    Topics Almost & η-Ricci solitons, Scalar curvature, Ricci solitons · Shared terms “Einstein”

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. 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.