Paper · Jun 15, 2019

From harmonic mappings to Ricci flow due to the Bochner technique

Sergey Stepanov, Irina Aleksandrova, Irina Tsyganok

math.DGarXiv:1906.07166
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Abstract

The present paper is devoted to the study a global aspect of the geometry of harmonic mappings and, in particular, infinitesimal harmonic transformations, and represents the application of our results to the theory of Ricci solutions and the Ricci flow. These results will be obtained using the methods of Geometric analysis and, in particular, due to theorems of Yau, Li and Schoen on the connections between the geometry of a complete smooth manifold and the global behavior of its subharmonic functions.

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Submitted
Jun 15, 2019
Subjects
math.DG

Comments: arXiv admin note: substantial text overlap with arXiv:1801.06433

Ricci flowHarmonic maps & DeTurck trick

Its neighbourhood

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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 On the stability of harmonic maps under the homogeneous Ricci flow (Rafaela F. do Prado and Lino Grama, 2017)7Prado 2017 From harmonic mappings to Ricci solitons (Sergey Stepanov and Irina Tsyganok, 2018)1Stepanov 2018 On the nonexistence of harmonic and bi-harmonic maps (Ahmed Mohammed Cherif, 2020)6Cherif 2020 From the Ricci flow evolution equation to vanishing theorems for Ricci solitons (Vladimir Rovenski, Sergey Stepanov and Irina Tsyganok, 2020)2Rovenski 2020 Short time existence for harmonic map heat flow with time-dependent metrics (Shaochuang Huang and Luen-Fai Tam, 2021)8Huang 2021 New applications of the Ahlfors Laplacian (Sergey E. Stepanov, Josef Mikes and Irina I. Tsyganok, 2023)5Stepanov 2023 Back to almost Ricci solitons (Vladimir Rovenski, Sergey Stepanov and Irina Tsyganok, 2023)3Rovenski 2023 The Harmonic Variational Principle for the Einstein-Hilbert Functional (Sergey Stepanov and Irina Tsyganok, 2026)4Stepanov 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 solitons

    Sergey Stepanov and Irina Tsyganok

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

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

    Vladimir Rovenski, Sergey Stepanov and Irina Tsyganok

    Topics Ricci flow · Same authors Sergey Stepanov, Irina Tsyganok

  3. Back to almost Ricci solitons

    Vladimir Rovenski, Sergey Stepanov and Irina Tsyganok

    Same authors Sergey Stepanov, Irina Tsyganok · Shared terms “geometric analysis”

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

    Sergey Stepanov and Irina Tsyganok

    Same authors Sergey Stepanov, Irina Tsyganok

  5. New applications of the Ahlfors Laplacian

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

    Same authors Sergey Stepanov, Irina Tsyganok

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

    Ahmed Mohammed Cherif

    Topics Harmonic maps & DeTurck trick

  7. On the stability of harmonic maps under the homogeneous Ricci flow

    Rafaela F. do Prado and Lino Grama

    Topics Harmonic maps & DeTurck trick

  8. Short time existence for harmonic map heat flow with time-dependent metrics

    Shaochuang Huang and Luen-Fai Tam

    Topics Harmonic maps & DeTurck trick

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 flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.