Skip to main content
SpringerLink
Log in

Coupled Sasaki-Ricci solitons

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

Motivated by the study of coupled Kähler-Einstein metrics by Hultgren and Witt Nyström (2018) and coupled Kähler-Ricci solitons by Hultgren (2017), we study in this paper coupled Sasaki-Einstein metrics and coupled Sasaki-Ricci solitons. We first show an isomorphism between the Lie algebra of all transverse holomorphic vector fields and certain space of coupled basic functions related to coupled twisted Laplacians for basic functions, and obtain extensions of the well-known obstructions to the existence of Kähler-Einstein metrics to this coupled case. These results are reduced to coupled Kähler-Einstein metrics when the Sasaki structure is regular. Secondly we show the existence of toric coupled Sasaki-Einstein metrics when the basic first Chern class is positive extending the work of Hultgren (2017).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aubin T. Réduction du cas positif de l’équation de Monge-Ampère sur les variétés Kählériennes compactes á la démonstration d’une inégalité. J Funct Anal, 1984, 57: 143–153

    Article  MathSciNet  Google Scholar 

  2. Boyer C P, Galicki K. Sasakian Geometry. Oxford Mathematical Monographs. Oxford: Oxford University Press, 2008

    Google Scholar 

  3. Cho K, Futaki A, Ono H. Uniqueness and examples of compact toric Sasaki-Einstein metrics. Comm Math Phys, 2008, 277: 439–458

    Article  MathSciNet  Google Scholar 

  4. Ding W Y. Remarks on the existence problem of positive Kähler-Einstein metrics. Math Ann, 1988, 282: 463–471

    Article  MathSciNet  Google Scholar 

  5. EL Kacimi-Alaoui A. Operateurs transversalement elliptiques sur un feuilletage riemannien et applications. Compos Math, 1990, 73: 57–106

    MathSciNet  MATH  Google Scholar 

  6. Futaki A. An obstruction to the existence of Einstein Kähler metrics. Invent Math, 1983, 73: 437–443

    Article  MathSciNet  Google Scholar 

  7. Futaki A. On a character of the automorphism group of a compact complex manifold. Invent Math, 1987, 87: 655–660

    Article  MathSciNet  Google Scholar 

  8. Futaki A. The Ricci curvature of symplectic quotients of Fano manifolds. Tohoku Math J (2), 1987, 39: 329–339

    Article  MathSciNet  Google Scholar 

  9. Futaki A. Kähler-Einstein Metrics and Integral Invariants. Lecture Notes in Mathematics, vol. 1314. Berlin: Springer-Verlag, 1988

    Google Scholar 

  10. Futaki A, Morita S. Invariant polynomials of the automorphism group of a compact complex manifold. J Differential Geom, 1985, 21: 135–142

    Article  MathSciNet  Google Scholar 

  11. Futaki A, Ono H. Cahen-Gutt moment map, closed Fedosov star product and structure of the automorphism group. J Symplectic Geom, 2019, in press

  12. Futaki A, Ono H. Volume minimization and obstructions to solving some problems in Kähler geometry. ICCM Notices, 2019, in press

  13. Futaki A, Ono H. Conformally Einstein-Maxwell Kähler metrics and the structure of the automorphisms. Math Z, 2019, 292: 571–589

    Article  MathSciNet  Google Scholar 

  14. Futaki A, Ono H, Wang G. Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds. J Differential Geom, 2009, 83: 585–635

    Article  MathSciNet  Google Scholar 

  15. Guillemin V. Moment Maps and Combinatorial Invariants of Hamiltonian T n-Spaces. Boston: Birkhäuser, 1994

    Book  Google Scholar 

  16. Hultgren J. Coupled Kähler-Ricci solitons on toric manifolds. ArXiv:1711.09881, 2017

  17. Hultgren J, Witt Nyström D. Coupled Kähler-Einstein metrics. Int Math Res Not IMRN, 2018, https://doi.org/10.1093/imrn/rnx298

  18. La Fuente-Gravy L. Futaki invariant for Fedosov’s star products. J Symplectic Geom, 2019, in press

  19. Lahdili A. Automorphisms and deformations of conformally Kähler, Einstein-Maxwell metrics. J Geom Anal, 2019, 29: 542–568

    Article  MathSciNet  Google Scholar 

  20. Legendre E. Toric geometry of convex quadrilaterals. J Symplectic Geom, 2011, 9: 343–385

    Article  MathSciNet  Google Scholar 

  21. Martelli D, Sparks J, Yau S-T. Sasaki-Einstein manifolds and volume minimisation. Comm Math Phys, 2008, 280: 611–673

    Article  MathSciNet  Google Scholar 

  22. Matsushima Y. Sur la structure du groupe d’homéomorphismes analytiques d’une certaine variété kaehlérinne. Nagoya Math J, 1957, 11: 145–150

    Article  MathSciNet  Google Scholar 

  23. Pingali V P. Existence of coupled Kähler-Einstein metrics using the continuity method. Internat J Math, 2018, 29: 1850041

    Article  MathSciNet  Google Scholar 

  24. Tian G, Zhu X-H. A new holomorphic invariant and uniqueness of Kähler-Ricci solitons. Comment Math Helv, 2002, 77: 297–325

    Article  MathSciNet  Google Scholar 

  25. Wang X-J, Zhu X. Kähler-Ricci solitons on toric manifolds with positive first Chern class. Adv Math, 2004, 188: 87–103

    Article  MathSciNet  Google Scholar 

  26. Yau S-T. On the Ricci curvature of a compact Kahler manifold and the complex Monge-Ampère equation I. Comm Pure Appl Math, 1978, 31: 339–441

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China

    Akito Futaki & Yingying Zhang

Authors
  1. Akito Futaki
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yingying Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Akito Futaki.

Additional information

In Memory of Professor Zhengguo Bai (1916–2015)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Futaki, A., Zhang, Y. Coupled Sasaki-Ricci solitons. Sci. China Math. 64, 1447–1462 (2021). https://doi.org/10.1007/s11425-018-9499-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-018-9499-y

Keywords

MSC(2010)

Search

Navigation