Paper · Oct 9, 2024

On the existence and classification of -Yamabe gradient solitons

Maria Fernanda Espinal, Mariel Sáez

math.DGarXiv:2410.06942v2
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Abstract

In this paper we classify rotationally symmetric conformally flat admissible solitons to the -Yamabe flow, a fully non-linear version of the Yamabe flow. For we prove existence of complete expanding, steady and shrinking solitons and describe their asymptotic behavior at infinity. For we prove that steady and expanding solitons are not admissible. The proof is based on the careful analysis of an associated dynamical system.

Yamabe flowRicci solitonsRotational symmetry & warped products

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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.

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 generalized Yamabe solitons under vanishing conditions on the Weyl, Cotton, and Cao-Chen tensors (Shun Maeta, 2021)8Maeta 2021 The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant (Gilles Carron, Eric Chen and Yi Wang, 2022)7Carron 2022 Existence of the (α,β)-Ricci-Yamabe flow on closed manifolds (Liangdi Zhang, 2023)2Zhang 2023 Some properties of hyperbolic Yamabe solitons (Adara M. Blaga and Cihan Özgür, 2023)3Blaga 2023 Generalized Yamabe Flows (Jørgen Olsen Lye, Boris Vertman and Mannaim Gennaro Vitti, 2024)6Lye 2024 Gap Theorem on locally conformally flat manifold (Ming Hsiao and Man-Chun Lee, 2025)4Hsiao 2025 Uncountably many non-rotationally symmetric type II ancient Yamabe flows on the sphere (Haixia Chen, Seunghyeok Kim and Monica Musso, 2026)1Chen 2026 A non-convergence phenomenon for the CR Yamabe flow (Claudio Afeltra, Andrea Pinamonti and Pak Tung Ho, 2026)5Afeltra 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Uncountably many non-rotationally symmetric type II ancient Yamabe flows on the sphere

    Haixia Chen, Seunghyeok Kim and Monica Musso PDF

    Topics Yamabe flow, Rotational symmetry & warped products · Shared terms “rotationally symmetric”, “conformally”

  2. Existence of the -Ricci-Yamabe flow on closed manifolds

    Liangdi Zhang PDF

    Topics Yamabe flow

  3. Some properties of hyperbolic Yamabe solitons

    Adara M. Blaga and Cihan ÖzgürWider flows PDF

    Topics Yamabe flow, Rotational symmetry & warped products

  4. Gap Theorem on locally conformally flat manifold

    Ming Hsiao and Man-Chun LeeWider flows PDF

    Topics Yamabe flow · Shared terms “conformally flat”

  5. A non-convergence phenomenon for the CR Yamabe flow

    Claudio Afeltra, Andrea Pinamonti and Pak Tung HoWider flows PDF

    Topics Yamabe flow

  6. Generalized Yamabe Flows

    Jørgen Olsen Lye, Boris Vertman and Mannaim Gennaro VittiWider flows PDF

    Topics Yamabe flow

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

    Gilles Carron, Eric Chen and Yi WangWider flows PDF

    Topics Yamabe flow

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

    Shun Maeta PDF

    Topics Rotational symmetry & warped products, Ricci solitons · Shared terms “gradient solitons”, “Yamabe”, “conformally flat”

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 within its conformal class, driven by scalar curvature. It is a parabolic approach to the Yamabe problem: find a… Start with: The yamabe flow on locally conformally flat manifolds with positive ricci curvature (Chow, 1992) 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) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

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. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  3. 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.