Paper · Jan 19, 2018

From harmonic mappings to Ricci solitons

Sergey Stepanov, Irina Tsyganok

math.DGarXiv:1801.06433
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Abstract

The paper is devoted to the study of the global geometries of harmonic mappings and infinitesimal harmonic transformations and presents their applications to the theory of Ricci solitons.

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Submitted
Jan 19, 2018
Subjects
math.DG
Ricci solitonsHarmonic maps & DeTurck trick

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 From harmonic mappings to Ricci flow due to the Bochner technique (Sergey Stepanov, Irina Aleksandrova and Irina Tsyganok, 2019)1Stepanov 2019 On the nonexistence of harmonic and bi-harmonic maps (Ahmed Mohammed Cherif, 2020)3Cherif 2020 From the Ricci flow evolution equation to vanishing theorems for Ricci solitons (Vladimir Rovenski, Sergey Stepanov and Irina Tsyganok, 2020)4Rovenski 2020 New applications of the Ahlfors Laplacian (Sergey E. Stepanov, Josef Mikes and Irina I. Tsyganok, 2023)6Stepanov 2023 Back to almost Ricci solitons (Vladimir Rovenski, Sergey Stepanov and Irina Tsyganok, 2023)5Rovenski 2023 On Ricci Solitons and Harmonic Vector Fields in the Thurston Geometry F⁴ (Halima Boukhari, Hadjer Okbani and Ahmed Mohammed Cherif, 2026)8Boukhari 2026 Ricci Solitons on the Lie Group Sol³_(m,n) and Applications (Mohammed El Amine Mekki and Ahmed Mohammed Cherif, 2026)7Mekki 2026 The Harmonic Variational Principle for the Einstein-Hilbert Functional (Sergey Stepanov and Irina Tsyganok, 2026)2Stepanov 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. From harmonic mappings to Ricci flow due to the Bochner technique

    Sergey Stepanov, Irina Aleksandrova and Irina Tsyganok

    Topics Harmonic maps & DeTurck trick · Same authors Sergey Stepanov, Irina Tsyganok · Shared terms “harmonic mappings”, “harmonic transformations”, “infinitesimal harmonic”

  2. The Harmonic Variational Principle for the Einstein-Hilbert Functional

    Sergey Stepanov and Irina Tsyganok

    Topics Ricci solitons · Same authors Sergey Stepanov, Irina Tsyganok

  3. On the nonexistence of harmonic and bi-harmonic maps

    Ahmed Mohammed Cherif

    Topics Harmonic maps & DeTurck trick, Ricci solitons

  4. From the Ricci flow evolution equation to vanishing theorems for Ricci solitons

    Vladimir Rovenski, Sergey Stepanov and Irina Tsyganok

    Topics Ricci solitons · Same authors Sergey Stepanov, Irina Tsyganok

  5. Back to almost Ricci solitons

    Vladimir Rovenski, Sergey Stepanov and Irina Tsyganok

    Topics Ricci solitons · Same authors Sergey Stepanov, Irina Tsyganok

  6. New applications of the Ahlfors Laplacian

    Sergey E. Stepanov, Josef Mikes and Irina I. Tsyganok

    Topics Ricci solitons · Same authors Sergey Stepanov, Irina Tsyganok

  7. Ricci Solitons on the Lie Group and Applications

    Mohammed El Amine Mekki and Ahmed Mohammed Cherif

    Topics Harmonic maps & DeTurck trick, Ricci solitons

  8. On Ricci Solitons and Harmonic Vector Fields in the Thurston Geometry

    Halima Boukhari, Hadjer Okbani and Ahmed Mohammed Cherif

    Topics Harmonic maps & DeTurck trick, 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) Harmonic maps & DeTurck trick Ricci flow is only weakly parabolic, because it is invariant under diffeomorphisms. The DeTurck trick adds a Lie-derivative term that makes it… Start with: Harmonic Mappings of Riemannian Manifolds (Eells et al., 1964)

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.