Paper · Nov 6, 2020

Rigidity results for quotient almost Yamabe solitons

Marcelo Bezerra Barboza, Willian Isao Tokura, Elismar Dias Batista, Priscila Marques Kai

math.DGWider flowsarXiv:2011.03569
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Abstract

In this paper we investigate the structure of certain solutions of the fully nonlinear Yamabe flow, which we call quotient almost Yamabe solitons because they extend quite naturally those called quotient Yamabe solitons. We then present sufficient conditions for a compact quotient almost Yamabe soliton to be either trivial or isometric with an Euclidean sphere. We also characterize noncompact quotient gradient almost Yamabe solitons satisfying certain conditions on both its Ricci tensor and potential function.

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Submitted
Nov 6, 2020
Subjects
math.DG
Yamabe flowAlmost & η-Ricci solitonsStability & uniquenessBoundaries & non-compact

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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 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 Classification of almost Yamabe solitons in Euclidean spaces (Tatsuya Seko and Shun Maeta, 2017)2Seko 2017 Convergence of the CR Yamabe Flow (Pak Tung Ho, Weimin Sheng and Kunbo Wang, 2017)6Ho 2017 Gradient Estimates on Warped Product Gradient Almost Ricci Solitons (Willian Isao Tokura, Levi Adriano, Romildo Pina et al., 2019)3Tokura 2019 CR Yamabe constant, CR Yamabe flow and its soliton (Pak Tung Ho and Kunbo Wang, 2020)1Ho 2020 Classification of generalized Yamabe solitons under vanishing conditions on the Weyl, Cotton, and Cao-Chen tensors (Shun Maeta, 2021)5Maeta 2021 The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant (Gilles Carron, Eric Chen and Yi Wang, 2022)7Carron 2022 Structure of generalized Yamabe solitons and its applications (Shun Maeta, 2022)8Maeta 2022 Some properties of hyperbolic Yamabe solitons (Adara M. Blaga and Cihan Özgür, 2023)4Blaga 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. CR Yamabe constant, CR Yamabe flow and its soliton

    Pak Tung Ho and Kunbo WangWider flows

    Topics Yamabe flow, Almost & η-Ricci solitons, Boundaries & non-compact

  2. Classification of almost Yamabe solitons in Euclidean spaces

    Tatsuya Seko and Shun Maeta

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

  3. Gradient Estimates on Warped Product Gradient Almost Ricci Solitons

    Willian Isao Tokura, Levi Adriano, Romildo Pina et al.

    Topics Almost & η-Ricci solitons · Same authors Marcelo Bezerra Barboza, Willian Isao Tokura · Shared terms “gradient almost”

  4. Some properties of hyperbolic Yamabe solitons

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

    Shun Maeta

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

  6. Convergence of the CR Yamabe Flow

    Pak Tung Ho, Weimin Sheng and Kunbo WangWider flows

    Topics Yamabe flow

  7. The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant

    Gilles Carron, Eric Chen and Yi WangWider flows

    Topics Yamabe flow, Boundaries & non-compact · Shared terms “Euclidean”

  8. Structure of generalized Yamabe solitons and its applications

    Shun Maeta

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

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.

Yamabe flow Wider flows The Yamabe flow deforms a metric inside its conformal class at a rate set by scalar curvature. It is a parabolic approach to the Yamabe problem… Start with: The yamabe flow on locally conformally flat manifolds with positive ricci curvature (Chow, 1992) Almost & η-Ricci solitons Variations on the Ricci soliton equation: let the constant λ become a function (almost solitons), add a term along a contact form (η-Ricci… Start with: Ricci almost solitons (Pigola et al., 2010) 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. 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.