Paper · Jun 24, 2018

Basic structural equations for almost Ricci-harmonic solitons and applications

Abimbola Abolarinwa

math.DGarXiv:1806.09190
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Abstract

This paper studies gradient almost Ricci-harmonic soliton with respect to a fixed metric. We rely on analytic techniques to estabilish some basic elliptic and integral equations for the structure of almost Ricci-harmonic soliton which generalizes that of Ricci-hamonic solitons on one hand and that of almost Ricci soliton on the other hand. The consequences of these formulas in relation to classification and rigidity results for gradient almost Ricci-harmonic solitons are also discussed.

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Submitted
Jun 24, 2018
Subjects
math.DG

Comments: 12 pages

Ricci solitonsAlmost & η-Ricci solitonsStability & uniqueness

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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 Almost η-Ricci solitons in (LCS)ₙ-manifolds (Adara M. Blaga, 2017)8Blaga 2017 The k-almost Ricci solitons and contact geometry (Amalendu Ghosh and Dhriti Sundar Patra, 2018)7Ghosh 2018 Harmonic aspects in an η-Ricci soliton (Adara M. Blaga, 2018)6Blaga 2018 On a Class of Gradient Almost Ricci Solitons (Sinem Güler, 2020)5Güler 2020 On conformal collineation and almost Ricci solitons (Adara M. Blaga and Bang-Yen Chen, 2022)4Blaga 2022 Back to almost Ricci solitons (Vladimir Rovenski, Sergey Stepanov and Irina Tsyganok, 2023)3Rovenski 2023 Almost Ricci solitons on Finsler spaces (Qiaoling Xia, 2024)1Xia 2024 Characterizations of Almost Ricci Bourguignon Solitons (Mohammad Aqib, Hemangi Madhusudan Shah and Dhriti Sundar Patra, 2025)2Aqib 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Almost Ricci solitons on Finsler spaces

    Qiaoling Xia

    Topics Almost & η-Ricci solitons, Stability & uniqueness, Ricci solitons · Shared terms “gradient almost”

  2. Characterizations of Almost Ricci Bourguignon Solitons

    Mohammad Aqib, Hemangi Madhusudan Shah and Dhriti Sundar Patra

    Topics Almost & η-Ricci solitons, Stability & uniqueness, Ricci solitons

  3. Back to almost Ricci solitons

    Vladimir Rovenski, Sergey Stepanov and Irina Tsyganok

    Topics Almost & η-Ricci solitons, Ricci solitons

  4. On conformal collineation and almost Ricci solitons

    Adara M. Blaga and Bang-Yen Chen

    Topics Almost & η-Ricci solitons, Ricci solitons

  5. On a Class of Gradient Almost Ricci Solitons

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

  6. Harmonic aspects in an -Ricci soliton

    Adara M. Blaga

    Topics Almost & η-Ricci solitons, Ricci solitons · Shared terms “Ricci harmonic”, “relation”

  7. The k-almost Ricci solitons and contact geometry

    Amalendu Ghosh and Dhriti Sundar Patra

    Topics Almost & η-Ricci solitons, Ricci solitons

  8. Almost -Ricci solitons in -manifolds

    Adara M. Blaga

    Topics Almost & η-Ricci solitons, Ricci solitons

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) 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)

In the guidebook

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

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

  2. 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.