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G-Sasaki Manifolds and K-Energy

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Abstract

In this paper, we introduce a class of Sasaki manifolds, called G-Sasaki manifolds with a reductive G-group action on their Kähler cones. By proving the properness of K-energy on such manifolds, we obtain a sufficient and necessary condition for the existence of G-Sasaki–Einstein metrics. A similar result is also obtained for G-Sasaki–Ricci solitons. As an application, we construct many new examples of G-Sasaki–Ricci solitons by an established openness theorem.

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Notes

  1. According to a recent work of He, Theorem 5.6 gives a criterion for the existence of transverse constant scalar curvature metrics on G-Sasaki manifolds [29].

  2. The authors would like to thank the referee for telling them the reference [30].

  3. Proposition 6.1 will be also used in the proof of Theorem 1.2 in Sect. 8.

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Acknowledgements

Y. Li was partially supported by BX20180010 and X. Zhu by NSFC Grants 11771019.

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  1. School of Mathematical Sciences and BICMR, Peking University, Beijing, 100871, China

    Yan Li & Xiaohua Zhu

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  1. Yan Li
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  2. Xiaohua Zhu
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Correspondence to Xiaohua Zhu.

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Li, Y., Zhu, X. G-Sasaki Manifolds and K-Energy. J Geom Anal 31, 1415–1470 (2021). https://doi.org/10.1007/s12220-019-00285-1

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