Paper · Apr 30, 2019

Gradient Estimates on Warped Product Gradient Almost Ricci Solitons

Willian Isao Tokura, Levi Adriano, Romildo Pina, Marcelo Barboza

math.DGarXiv:1905.00068
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Abstract

In this paper, by slightly modifying Li-Yau's technique so that we can handle drifting Laplacians, we were able to find three different gradient estimates for the warping function, one for each sign of the Einstein constant of the fiber manifold. As an application, we exhibit a nonexistence theorem for gradient almost Ricci solitons possessing certain metric properties on the base of the warped product.

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Submitted
Apr 30, 2019
Subjects
math.DG
Ricci solitonsAlmost & η-Ricci solitonsHarnack inequalitiesRotational symmetry & warped products

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 Classification of base of warped product almost Ricci solitons (José N. V. Gomes and Manoel V. M. Neto, 2017)3Gomes 2017 On warped product gradient η-Ricci solitons (Adara M. Blaga, 2017)1Blaga 2017 On a Class of Gradient Almost Ricci Solitons (Sinem Güler, 2020)2Güler 2020 On the construction of complete expanding gradient Rici solitons (Marcio de Sousa, Paula Bonfim, Romildo Pina et al., 2020)6de Sousa 2020 Rigidity results for quotient almost Yamabe solitons (Marcelo Bezerra Barboza, Willian Isao Tokura, Elismar Dias Batista et al., 2020)4Barboza 2020 On noncompact warped product Ricci solitons (Valter Borges, 2022)7Borges 2022 Local structure of gradient almost Ricci solitons with harmonic Weyl tensor (Valter Borges, Matheus Andrade Ribeiro de Moura Horácio and João Paulo dos Santos, 2025)8Borges 2025 Gradient estimates for a parabolic partial differential equation under the Ricci-Bourguignon flow (José N. V. Gomes, Willian I. Tokura and Hikaru Yamamoto, 2026)5Gomes 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On warped product gradient -Ricci solitons

    Adara M. Blaga

    Topics Almost & η-Ricci solitons, Rotational symmetry & warped products, Ricci solitons · Shared terms “product gradient”, “Laplacian”

  2. On a Class of Gradient Almost Ricci Solitons

    Sinem Güler

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

  3. Classification of base of warped product almost Ricci solitons

    José N. V. Gomes and Manoel V. M. Neto

    Topics Almost & η-Ricci solitons, Rotational symmetry & warped products, Ricci solitons

  4. Rigidity results for quotient almost Yamabe solitons

    Marcelo Bezerra Barboza, Willian Isao Tokura, Elismar Dias Batista et al.Wider flows

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

  5. Gradient estimates for a parabolic partial differential equation under the Ricci-Bourguignon flow

    José N. V. Gomes, Willian I. Tokura and Hikaru Yamamoto

    Topics Rotational symmetry & warped products · Same author Willian Isao Tokura · Shared terms “gradient estimates”, “warping function”

  6. On the construction of complete expanding gradient Rici solitons

    Marcio de Sousa, Paula Bonfim, Romildo Pina et al.

    Topics Rotational symmetry & warped products, Ricci solitons · Same author Romildo Pina · Shared terms “warping function”

  7. On noncompact warped product Ricci solitons

    Valter Borges

    Topics Rotational symmetry & warped products, Ricci solitons · Shared terms “product gradient”, “nonexistence theorem”, “warping function”

  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

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

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) 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) Harnack inequalities Differential Harnack inequalities compare a solution with itself at different points and times. Hamilton's matrix Harnack inequality says curvature… Start with: On the parabolic kernel of the Schrödinger operator (Li et al., 1986) 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. 6D.5
    Diffusions on Manifolds and CurvatureBook 6D · Diffusions, Finance and Computation

    Brownian motion on a Riemannian manifold, the Bakry–Émery calculus, curvature-dimension inequalities, gradient estimates and log-Sobolev, coupling, and Brownian motion along a Ricci flow.

  2. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

  3. 12A.6
    𝓛-Geodesics and the Variational FormulasBook 12A · Entropy and Noncollapsing

    Perelman's §7.1 derived line by line: the first and second variation of 𝓛-length, 𝓛-Jacobi fields, adapted fields, Hamilton's Harnack expressions, and the differential inequalities for the reduced distance.