Paper · Jul 19, 2019

A conformal Yamabe problem with potential on the euclidean space

Giovanni Catino, Filippo Gazzola, Paolo Mastrolia

math.DGarXiv:1907.08476v2
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Abstract

We consider, in the Euclidean setting, a conformal Yamabe-type equation related to a potential generalization of the classical constant scalar curvature problem and which naturally arises in the study of Ricci solitons structures. We prove existence and nonexistence results, focusing on the radial case, under some general hypothesis on the potential.

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Submitted
Jul 19, 2019
Last revised
Nov 13, 2019 (v2)
Subjects
math.DG, math.AP
Ricci solitonsScalar curvature

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 A potential generalization of some canonical Riemannian metrics (Giovanni Catino and Paolo Mastrolia, 2017)1Catino 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 On Ricci solitons whose potential is convex (Chandan Kumar Mondal and Absos Ali Shaikh, 2019)7Mondal 2019 Classification of generalized Yamabe solitons under vanishing conditions on the Weyl, Cotton, and Cao-Chen tensors (Shun Maeta, 2021)5Maeta 2021 Gradient Ricci solitons carrying a closed conformal vector field (J. F. Siva Filho and R. Sharma, 2021)8Filho 2021 Rigidity of shrinking gradient ricci soliton with constant scalar curvature (Fengjiang Li, Yuanyuan Qu and Guoqiang Wu, 2026)6Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A potential generalization of some canonical Riemannian metrics

    Giovanni Catino 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

  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

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

    Giovanni Catino

    Topics Scalar curvature, Ricci solitons · Same author Giovanni Catino

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

    Shun Maeta

    Topics Scalar curvature, Ricci solitons · Shared terms “Yamabe type”

  6. Rigidity of shrinking gradient ricci soliton with constant scalar curvature

    Fengjiang Li, Yuanyuan Qu and Guoqiang Wu

    Topics Scalar curvature, Ricci solitons · Shared terms “constant scalar”

  7. On Ricci solitons whose potential is convex

    Chandan Kumar Mondal and Absos Ali Shaikh

    Topics Scalar curvature, Ricci solitons

  8. Gradient Ricci solitons carrying a closed conformal vector field

    J. F. Siva Filho and R. Sharma

    Topics Scalar curvature, Ricci solitons · Shared terms “constant scalar”, “Euclidean space”

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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

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.