Paper · Oct 28, 2019

Sasaki-Ricci flow equation on five-dimensional Sasaki-Einstein space

Mihai Visinescu

hep-tharXiv:1910.12495
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Abstract

We analyze the transverse Kähler-Ricci flow equation on Sasaki-Ein\-stein space . Explicit solutions are produced representing new five-dimensional Sasaki structures. Solutions which do not modify the transverse metric preserve the Sasaki-Einstein feature of the contact structure. If the transverse metric is altered, the deformed metrics remain Sasaki, but not Einstein.

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Submitted
Oct 28, 2019
Subjects
hep-th
Journal
Mod. Phys. Lett. A 35 (2020) 2050114
DOI
10.1142/S021773232050114X

Comments: 12 pages

Ricci flowKähler–Ricci flowEinstein metricsKähler geometrySasakian & contact geometry

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

    Vladimir Slesar, Mihai Visinescu and Gabriel-Eduard Vîlcu

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

  2. Transverse Rigidity of Shrinking Sasaki-Ricci Solitons

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

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

  3. -Sasaki manifolds and K-energy

    Yan Li and Xiaohua Zhu

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

  4. Coupled Sasaki-Ricci solitons

    Akito Futaki and Yingying Zhang

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

  5. 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 Einstein”, “Sasaki Ricci”, “transverse”

  6. 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 Ricci”, “transverse”

  7. Log del Pezzo -surfaces, Kähler-Einstein metrics, Kähler-Ricci solitons and Sasaki-Einstein metrics

    Daniel Hättig, Jürgen Hausen and Hendrik Süß

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

  8. Sasaki with torsion manifolds and string backgrounds

    Beatrice Brienza, Anna Fino, Udhav Fowdar et al.

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

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.