Paper · May 30, 2017

A potential generalization of some canonical Riemannian metrics

Giovanni Catino, Paolo Mastrolia

math.DGarXiv:1705.10599
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Abstract

The aim of this paper is to study new classes of Riemannian manifolds endowed with a smooth potential function, including in a general framework classical canonical structures such as Einstein, harmonic curvature and Yamabe metrics, and, above all, gradient Ricci solitons. For the most rigid cases we give a complete classification, while for the others we provide rigidity and obstruction results, characterizations and nontrivial examples. In the final part of the paper we also describe the "nongradient" version of this construction.

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Submitted
May 30, 2017
Subjects
math.DG
Journal
Ann. Glob. Anal. Geom. 55 (2019), no. 4, 719-748
Ricci solitonsStability & uniqueness

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 Gradient Ricci solitons with vanishing conditions on Weyl (Giovanni Catino, Paolo Mastrolia and Dario Daniele Monticelli, 2016)3Catino 2016 On a classification of 4-d gradient Ricci solitons with harmonic Weyl curvature (Jongsu Kim, 2016)7Kim 2016 A new approach to gradient Ricci solitons and generalizations (Mircea Crasmareanu, 2017)5Crasmareanu 2017 Weyl scalars on compact Ricci solitons (Giovanni Catino and Paolo Mastrolia, 2018)2Catino 2018 Rigidity of positively curved shrinking Ricci solitons in dimension four (Giovanni Catino, 2018)4Catino 2018 A conformal Yamabe problem with potential on the euclidean space (Giovanni Catino, Filippo Gazzola and Paolo Mastrolia, 2019)1Catino 2019 On the rigidity of harmonic-Ricci solitons (Andrea Anselli, 2020)6Anselli 2020 Almost Ricci solitons on Finsler spaces (Qiaoling Xia, 2024)8Xia 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A conformal Yamabe problem with potential on the euclidean space

    Giovanni Catino, Filippo Gazzola and Paolo Mastrolia

    Topics Ricci solitons · Same authors Giovanni Catino, Paolo Mastrolia · Shared terms “potential generalization”, “Yamabe”

  2. Weyl scalars on compact Ricci solitons

    Giovanni Catino and Paolo Mastrolia

    Topics Ricci solitons · Same authors Giovanni Catino, Paolo Mastrolia · Shared terms “potential function”

  3. Gradient Ricci solitons with vanishing conditions on Weyl

    Giovanni Catino, Paolo Mastrolia and Dario Daniele Monticelli

    Topics Ricci solitons · Same authors Giovanni Catino, Paolo Mastrolia · Shared terms “gradient Ricci”

  4. Rigidity of positively curved shrinking Ricci solitons in dimension four

    Giovanni Catino

    Topics Stability & uniqueness, Ricci solitons · Same author Giovanni Catino

  5. A new approach to gradient Ricci solitons and generalizations

    Mircea Crasmareanu

    Topics Ricci solitons · Shared terms “manifolds endowed”, “gradient Ricci”

  6. On the rigidity of harmonic-Ricci solitons

    Andrea Anselli

    Topics Stability & uniqueness, Ricci solitons · Shared terms “characterization”, “Riemannian manifold”

  7. On a classification of 4-d gradient Ricci solitons with harmonic Weyl curvature

    Jongsu Kim

    Topics Stability & uniqueness, Ricci solitons · Shared terms “harmonic curvature”, “gradient Ricci”, “characterization”

  8. Almost Ricci solitons on Finsler spaces

    Qiaoling Xia

    Topics Stability & uniqueness, Ricci solitons · Shared terms “nontrivial examples”, “gradient Ricci”, “characterization”

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)

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.