Paper · Mar 27, 2026

On a special class of gradient Ricci solitons

José Nazareno Vieira Gomes, Marcus Antonio Mendonça Marrocos

math.DGarXiv:2603.26903v2
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Abstract

We develop a method for constructing complete gradient Ricci solitons realized as fiber bundles endowed with warped metrics, and we establish necessary and sufficient conditions for their existence. As an application, we present new examples of complete gradient steady and shrinking Ricci solitons obtained via quotients by isometric group actions

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Ricci solitons

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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 A note on gradient Ricci soliton warped metrics (José N. V. Gomes, Marcus A. M. Marrocos and Adrian V. C. Ribeiro, 2021)1Gomes 2021 Mean curvature flow in an extended Ricci flow background (José N. V. Gomes and Matheus Hudson, 2023)7Gomes 2023 Hadamard-type variation formulae for the eigenvalues of a class of second-order elliptic operators and its applications (Cleiton Lira Cunha, José Nazareno Vieira Gomes and Marcus Antônio Mendonça Marrocos, 2023)2Cunha 2023 Liouville theorems for harmonic 1-forms on gradient Ricci solitons (Chenghong He, Di Wu and Xi Zhang, 2024)3He 2024 A note on Huisken monotonicity-type formula for the mean curvature flow in a gradient shrinking extended Ricci soliton background (José N. V. Gomes, Matheus Hudson and Hikaru Yamamoto, 2024)6Gomes 2024 Mean curvature flow into an ambient Riemannian manifold evolving by Ricci flow coupled with harmonic map heat flow (José N. V. Gomes, Matheus Hudson and Carlos M. de Sousa, 2025)8Gomes 2025 Gradient estimates for a parabolic partial differential equation under the Ricci-Bourguignon flow (José N. V. Gomes, Willian I. Tokura and Hikaru Yamamoto, 2026)4Gomes 2026 Complete gradient Ricci solitons with zero radial Weyl curvature (Tongzhu Li and Junlong Yu, 2026)5Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A note on gradient Ricci soliton warped metrics

    José N. V. Gomes, Marcus A. M. Marrocos and Adrian V. C. Ribeiro PDF

    Topics Ricci solitons · Same authors José Nazareno Vieira Gomes, Marcus Antonio Mendonça Marrocos · Shared terms “warped metrics”, “gradient Ricci”, “realized”

  2. Hadamard-type variation formulae for the eigenvalues of a class of second-order elliptic operators and its applications

    Cleiton Lira Cunha, José Nazareno Vieira Gomes and Marcus Antônio Mendonça Marrocos PDF

    Same authors José Nazareno Vieira Gomes, Marcus Antonio Mendonça Marrocos

  3. Liouville theorems for harmonic 1-forms on gradient Ricci solitons

    Chenghong He, Di Wu and Xi Zhang PDF

    Topics Ricci solitons · Shared terms “complete gradient”, “gradient Ricci”, “gradient steady”

  4. Gradient estimates for a parabolic partial differential equation under the Ricci-Bourguignon flow

    José N. V. Gomes, Willian I. Tokura and Hikaru Yamamoto PDF

    Same author José Nazareno Vieira Gomes · Shared terms “sufficient conditions”, “constructing”, “necessary”

  5. Complete gradient Ricci solitons with zero radial Weyl curvature

    Tongzhu Li and Junlong Yu PDF

    Topics Ricci solitons · Shared terms “complete gradient”, “gradient Ricci”

  6. A note on Huisken monotonicity-type formula for the mean curvature flow in a gradient shrinking extended Ricci soliton background

    José N. V. Gomes, Matheus Hudson and Hikaru Yamamoto PDF

    Topics Ricci solitons · Same author José Nazareno Vieira Gomes

  7. Mean curvature flow in an extended Ricci flow background

    José N. V. Gomes and Matheus Hudson PDF

    Topics Ricci solitons · Same author José Nazareno Vieira Gomes · Shared terms “constructing”

  8. Mean curvature flow into an ambient Riemannian manifold evolving by Ricci flow coupled with harmonic map heat flow

    José N. V. Gomes, Matheus Hudson and Carlos M. de Sousa PDF

    Topics Ricci solitons · Same author José Nazareno Vieira Gomes · Shared terms “gradient steady”

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)

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.