Paper · May 22, 2020

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

Vladimir Rovenski, Sergey Stepanov, Irina Tsyganok

math.DGarXiv:2005.11392v2
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Abstract

In the paper, we study evolution equations of the scalar and Ricci curvatures under the Hamilton's Ricci flow on a closed manifold and on a complete noncompact manifold. In particular, we study conditions when the Ricci flow is trivial and the Ricci soliton is Ricci flat or Einstein.

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Submitted
May 22, 2020
Last revised
Sep 16, 2020 (v2)
Subjects
math.DG

Comments: 10 pages

Ricci flowRicci solitonsRicci-flat & Calabi–YauBoundaries & non-compact

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 solitons (Sergey Stepanov and Irina Tsyganok, 2018)5Stepanov 2018 From harmonic mappings to Ricci flow due to the Bochner technique (Sergey Stepanov, Irina Aleksandrova and Irina Tsyganok, 2019)7Stepanov 2019 On Complete Gradient Steady Ricci Solitons with Vanishing D-tensor (Huai-Dong Cao and Jiangtao Yu, 2020)1Cao 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)2Rovenski 2023 L^2-instability of the Taub-Bolt metric under the Ricci flow (John Hughes, 2024)8Hughes 2024 The Harmonic Variational Principle for the Einstein-Hilbert Functional (Sergey Stepanov and Irina Tsyganok, 2026)4Stepanov 2026 Conjugate radius and parabolicity along the Ricci flow (Chandan Kumar Mondal, 2026)3Mondal 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On Complete Gradient Steady Ricci Solitons with Vanishing D-tensor

    Huai-Dong Cao and Jiangtao Yu

    Topics Ricci-flat & Calabi–Yau, Boundaries & non-compact, Ricci solitons · Shared terms “vanishing”, “complete noncompact”

  2. Back to almost Ricci solitons

    Vladimir Rovenski, Sergey Stepanov and Irina Tsyganok

    Topics Ricci solitons · Same authors Vladimir Rovenski, Sergey Stepanov, Irina Tsyganok

  3. Conjugate radius and parabolicity along the Ricci flow

    Chandan Kumar Mondal

    Topics Ricci-flat & Calabi–Yau, Boundaries & non-compact, Ricci solitons · Shared terms “complete noncompact”, “noncompact manifolds”

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

    Sergey Stepanov and Irina Tsyganok

    Topics Ricci solitons · Same authors Sergey Stepanov, Irina Tsyganok

  5. From harmonic mappings to Ricci solitons

    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. From harmonic mappings to Ricci flow due to the Bochner technique

    Sergey Stepanov, Irina Aleksandrova and Irina Tsyganok

    Topics Ricci flow · Same authors Sergey Stepanov, Irina Tsyganok

  8. L^2-instability of the Taub-Bolt metric under the Ricci flow

    John Hughes

    Topics Ricci-flat & Calabi–Yau, Boundaries & non-compact, 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 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) 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) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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.

  3. 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.