Paper · Jun 14, 2020

On the rigidity of harmonic-Ricci solitons

Andrea Anselli

math.DGarXiv:2006.07892
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Abstract

In this paper we introduce the notion of rigidity for harmonic-Ricci solitons and we provide some characterizations of rigidity, generalizing some known results for Ricci solitons. In the compact case we are able to deal with not necessarily gradient solitons while, in the complete non-compact case, we restrict our attention to steady and shrinking gradient solitons. We show that the rigidity can be traced back to the vanishing of certain modified curvature tensors that take into account the geometry a Riemannian manifold equipped with a smooth map , called -curvatures, which are a natural generalization of the standard curvature tensors in the setting of harmonic-Ricci solitons.

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Submitted
Jun 14, 2020
Subjects
math.DG
Ricci solitonsStability & uniquenessBoundaries & non-compact

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 2016, 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 Ridigity of Ricci Solitons with Weakly Harmonic Weyl Tensors (Seungsu Hwang and Gabjin Yun, 2016)2Hwang 2016 Analysis of Type I Singularities in the Harmonic Ricci Flow (Gianmichele Di Matteo, 2018)3Di Matteo 2018 Rigidity of Complete Gradient Steady Ricci Solitons with Harmonic Weyl Curvature (Fengjiang Li, 2021)7Li 2021 Rigidity results for shrinking and expanding Ricci solitons (Benedito Leandro and Jeferson Poveda, 2022)5Leandro 2022 A note on Rigidity of Shrinking Gradient Ricci Solitons with Constant Scalar Curvature (Chen Wang and Guoqiang Wu, 2026)6Wang 2026 Notes on harmonic-Ricci flow on surface (Xiang-Zhi Cao, 2026)1Cao 2026 Uniqueness of shrinking Kähler-Ricci solitons on resolutions of Kähler cones (Ronan J. Conlon and Alix Deruelle, 2026)8Conlon 2026 Correspondence between Mean Curvature Flow and Harmonic-Ricci Flow (Tianyin Ren and Chong Song, 2026)4Ren 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Notes on harmonic-Ricci flow on surface

    Xiang-Zhi Cao

    Shared terms “harmonic Ricci”

  2. Ridigity of Ricci Solitons with Weakly Harmonic Weyl Tensors

    Seungsu Hwang and Gabjin Yun

    Topics Boundaries & non-compact, Stability & uniqueness, Ricci solitons · Shared terms “curvature tensor”

  3. Analysis of Type I Singularities in the Harmonic Ricci Flow

    Gianmichele Di Matteo

    Topics Ricci solitons · Shared terms “harmonic Ricci”, “generalizing”

  4. Correspondence between Mean Curvature Flow and Harmonic-Ricci Flow

    Tianyin Ren and Chong Song

    Shared terms “harmonic Ricci”

  5. Rigidity results for shrinking and expanding Ricci solitons

    Benedito Leandro and Jeferson Poveda

    Topics Boundaries & non-compact, Stability & uniqueness, Ricci solitons

  6. A note on Rigidity of Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

    Chen Wang and Guoqiang Wu

    Topics Boundaries & non-compact, Stability & uniqueness, Ricci solitons · Shared terms “shrinking gradient”, “vanishing”

  7. Rigidity of Complete Gradient Steady Ricci Solitons with Harmonic Weyl Curvature

    Fengjiang Li

    Topics Boundaries & non-compact, Stability & uniqueness, Ricci solitons · Shared terms “necessarily”

  8. Uniqueness of shrinking Kähler-Ricci solitons on resolutions of Kähler cones

    Ronan J. Conlon and Alix Deruelle

    Topics Boundaries & non-compact, Stability & uniqueness, Ricci solitons · Shared terms “shrinking gradient”, “necessarily”

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) 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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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.