Paper · May 13, 2019

Transverse Kähler-Ricci flow and deformations of the metric on the Sasaki space

Vladimir Slesar, Mihai Visinescu, Gabriel-Eduard Vîlcu

math-pharXiv:1905.05024v2
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Abstract

In this paper we investigate the possibility to obtain locally new Sasaki-Einstein metrics on the space considering a deformation of the standard metric tensor field. We show that from the geometric point of view this deformation leaves transverse and the leafwise metric intact, but changes the orthogonal complement of the Reeb vector field using a particular basic function. In particular, the family of metric obtained using this method can be regarded as solutions of the equation associated to the Sasaki-Ricci flow on the underlying manifold.

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Submitted
May 13, 2019
Last revised
Jun 20, 2019 (v2)
Subjects
math-ph, hep-th, math.DG
Journal
Rom. Rep. Phys. 72 (2020) 108

Comments: 19 pages, references added, typos fixed

Ricci flowKähler–Ricci flowEinstein metricsKähler geometrySasakian & contact geometry

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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 G-Sasaki manifolds and K-energy (Yan Li and Xiaohua Zhu, 2017)4Li 2017 Coupled Sasaki-Ricci solitons (Akito Futaki and Yingying Zhang, 2018)3Futaki 2018 Sasaki-Ricci flow equation on five-dimensional Sasaki-Einstein space Y^(p,q) (Mihai Visinescu, 2019)1Visinescu 2019 Geometry of almost contact metrics as almost *-Ricci solitons (Dhriti Sundar Patra, Akram Ali and Fatemah Mofarreh, 2021)8Patra 2021 Convergence of the Sasaki-Ricci flow on Sasakian 5-manifolds of general type (Shu-Cheng Chang, Yingbo Han, Chien Lin et al., 2022)7Chang 2022 Vaisman manifolds and transversally Kähler-Einstein metrics (Vladimir Slesar and Gabriel-Eduard Vîlcu, 2022)2Slesar 2022 On the existence of conic Sasaki-Einstein metrics on log Fano Sasakian manifolds of dimension five (Shu-Cheng Chang, Fengjiang Li, Chien Lin et al., 2024)6Chang 2024 Transverse Rigidity of Shrinking Sasaki-Ricci Solitons (Shu-Cheng Chang, Fengjiang Li, Chien Lin et al., 2025)5Chang 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Sasaki-Ricci flow equation on five-dimensional Sasaki-Einstein space

    Mihai Visinescu

    Topics Sasakian & contact geometry, Kähler–Ricci flow, Einstein metrics · Same author Mihai Visinescu · Shared terms “Sasaki”, “transverse Kähler”

  2. Vaisman manifolds and transversally Kähler-Einstein metrics

    Vladimir Slesar and Gabriel-Eduard Vîlcu

    Topics Kähler–Ricci flow, Einstein metrics, Kähler geometry · Same authors Vladimir Slesar, Gabriel-Eduard Vîlcu · Shared terms “transverse Kähler”

  3. Coupled Sasaki-Ricci solitons

    Akito Futaki and Yingying Zhang

    Topics Sasakian & contact geometry, Einstein metrics, Kähler geometry · Shared terms “Sasaki”, “basic functions”, “transverse”

  4. -Sasaki manifolds and K-energy

    Yan Li and Xiaohua Zhu

    Topics Sasakian & contact geometry, Einstein metrics · Shared terms “Sasaki”, “transverse”

  5. Transverse Rigidity of Shrinking Sasaki-Ricci Solitons

    Shu-Cheng Chang, Fengjiang Li, Chien Lin et al.

    Topics Sasakian & contact geometry · Shared terms “Sasaki”, “transverse”

  6. On the existence of conic Sasaki-Einstein metrics on log Fano Sasakian manifolds of dimension five

    Shu-Cheng Chang, Fengjiang Li, Chien Lin et al.

    Topics Sasakian & contact geometry, Einstein metrics, Kähler geometry · Shared terms “Sasaki”, “transverse”

  7. Convergence of the Sasaki-Ricci flow on Sasakian 5-manifolds of general type

    Shu-Cheng Chang, Yingbo Han, Chien Lin et al.

    Topics Sasakian & contact geometry, Einstein metrics · Shared terms “Sasaki”, “transverse”

  8. Geometry of almost contact metrics as almost -Ricci solitons

    Dhriti Sundar Patra, Akram Ali and Fatemah Mofarreh

    Topics Sasakian & contact geometry, Einstein metrics · Shared terms “Reeb vector”, “vector field”, “considering”

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) Kähler–Ricci flow Ricci flow started from a Kähler metric stays Kähler and reduces to a single scalar equation, a parabolic complex Monge–Ampère equation. Start with: Deformation of Kähler matrics to Kähler-Einstein metrics on compact Kähler manifolds (Cao, 1985) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

  1. 5A.6
    Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry

    The Kähler–Ricci flow, and when to come back to it.

  2. 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.

  3. 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.