Paper · Jul 12, 2021

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

Shun Maeta

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Abstract

We study complete conformal gradient solitons, a class containing gradient Yamabe solitons and many generalized Yamabe-type structures, including gradient almost Yamabe, gradient k-Yamabe, and gradient h-almost Yamabe solitons, and, after a change of the potential function, gradient Einstein-type manifolds with and (in particular, quasi-Yamabe solitons). In this paper, we classify complete nontrivial locally conformally flat conformal gradient solitons. This result contributes to an analogue of Perelman's conjecture for Yamabe-type solitons. Moreover, we show that under nonnegative scalar curvature, every nonflat soliton is rotationally symmetric. We also obtain classifications assuming the Cotton or Cao-Chen tensor vanishes.

Comments: 13 pages

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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.

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  1. Structure of generalized Yamabe solitons and its applications

    Shun Maeta PDF

    Topics Almost & η-Ricci solitons, Ricci solitons · Same author Shun Maeta · Shared terms “generalized Yamabe”, “Yamabe soliton”, “conformal gradient”

  2. Classification of gradient Einstein-type Kähler manifolds with

    Shun Maeta PDF

    Topics Almost & η-Ricci solitons, Rotational symmetry & warped products, Ricci solitons · Same author Shun Maeta · Shared terms “Yamabe soliton”, “gradient Einstein”, “Einstein type”

  3. Complete gradient Einstein-type Sasakian manifolds with

    Shun Maeta PDF

    Topics Almost & η-Ricci solitons, Ricci solitons · Same author Shun Maeta · Shared terms “gradient Einstein”, “Einstein type”, “Yamabe soliton”

  4. Solitons on Statistical Manifolds and Their Submanifolds

    Aliya Naaz Siddiqui and Bang-Yen Chen PDF

    Topics Almost & η-Ricci solitons, Rotational symmetry & warped products, Ricci solitons · Shared terms “Yamabe soliton”, “Chen”, “quasi Yamabe”

  5. Investigations on a Riemannian manifold with a semi-symmetric non-metric connection and gradient solitons

    Krishnendu De, Uday Chand De and Aydin Gezer PDF

    Topics Almost & η-Ricci solitons, Scalar curvature, Ricci solitons · Shared terms “Yamabe soliton”, “gradient solitons”, “gradient Yamabe”

  6. Ricci-Yamabe solitons on the Lie group

    Abdou Bousso and Ameth Ndiaye PDF

    Topics Almost & η-Ricci solitons, Ricci solitons · Shared terms “Yamabe soliton”, “Perelman”

  7. Some properties of hyperbolic Yamabe solitons

    Adara M. Blaga and Cihan ÖzgürWider flows PDF

    Topics Almost & η-Ricci solitons, Rotational symmetry & warped products · Shared terms “Yamabe soliton”

  8. 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, Rotational symmetry & warped products, Ricci solitons · Shared terms “gradient almost”, “Weyl”, “potential function”

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.

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.