Paper · Dec 21, 2017

-Sasaki manifolds and K-energy

Yan Li, Xiaohua Zhu

math.DGarXiv:1712.07934v2
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Abstract

In this paper, we introduce a class of Sasaki manifolds with a reductive -group action, called -Sasaki manifolds. By reducing K-energy to a functional defined on a class of convex functions on a moment polytope, we give a criterion for the properness of K-energy. In particular, we deduce a sufficient and necessary condition related to the polytope for the existence of transverse -Sasaki Einstein metrics. A similar result is also obtained for transverse -Sasaki Ricci solitons. As an application, we construct several examples of -Sasaki Ricci solitons by an established openness theorem for transverse -Sasaki Ricci solitons.

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Submitted
Dec 21, 2017
Last revised
Jan 16, 2018 (v2)
Subjects
math.DG

Comments: some typos were corrected and some references were added

Ricci solitonsEinstein metricsSasakian & contact geometry

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  1. Transverse Rigidity of Shrinking Sasaki-Ricci Solitons

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

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

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

    Mihai Visinescu

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

  3. Coupled Sasaki-Ricci solitons

    Akito Futaki and Yingying Zhang

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

  4. 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, Einstein metrics · Shared terms “Sasaki”, “transverse”

  5. 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”

  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, Ricci solitons · Shared terms “Sasaki Ricci”, “Sasaki Einstein”, “transverse”

  7. Gradient Shrinking Sasaki-Ricci Solitons with Harmonic Weyl Tensor

    Shu-Cheng Chang and Hongbing Qiu

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

  8. The weighted Hermite–Einstein equation

    Michael Hallam and Abdellah Lahdili

    Topics Sasakian & contact geometry, Einstein metrics, Ricci solitons · Shared terms “Sasaki manifolds”, “moment polytope”

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) 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) Sasakian & contact geometry Sasakian manifolds are odd-dimensional manifolds whose metric cone is Kähler. They carry a transverse Kähler structure, so Kähler–Ricci flow has an… Start with: THE SASAKI–RICCI FLOW (SMOCZYK et al., 2010)

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.