Skip to main content
SpringerLink
Log in

Uniqueness of constant scalar curvature Kähler metrics with cone singularities. I: reductivity

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

The aim of this paper is to investigate uniqueness of conic constant scalar curvature Kähler (cscK) metrics, when the cone angle is less than \(\pi \). We introduce a new Hölder space called \({\mathcal {C}}^{4,\alpha ,\beta }\) to study the regularities of this fourth order elliptic equation, and prove that any \({\mathcal {C}}^{2,\alpha ,\beta }\) conic cscK metric is indeed of class \({\mathcal {C}}^{4,\alpha ,\beta }\). Finally, the reductivity is established by a careful study of the conic Lichnerowicz operator.

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. Brendle, S.: Ricci flat Kähler metrics with edge singularities. Int. Math. Res. Not. IMRN 24, 5727–5766 (2013)

    Article  MATH  Google Scholar 

  2. Bando, S., Mabuchi, T.: Uniqueness of Einstein Kähler metrics modulo connected group actions, Algebraic geometry, Sendai, 1985. Advanced Studies in Pure Mathematics, vol. 10. North-Holland, Amsterdam, pp. 11–40 (1987)

  3. Berman, R., Berndtsson, B.: Convexity of the K-energy on the space of Kähler metrics. arXiv:1405.0401

  4. Berndtsson, B.: A Brunn–Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry. Invent. Math. 200(1), 149–200 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Berman, R., Boucksom, S., Eyssidieux, P., Guedj, V., Zeriahi, A.: Kähler-Einstein metrics and the Kähler–Ricci flow on log Fano varieties. arXiv:1111.7158

  6. Calamai, S., Zheng, K.: Geodesics in the space of Kähler cone metrics, I. Am. J. Math. 137(5), 1149–1208 (2015)

    Article  MATH  Google Scholar 

  7. Chen, X.: On the existence of constant scalar curvature Kähler metric: a new perspective. arXiv:1506.06423

  8. Chen, X.X., Li, L., Păun, M.: Approximation of weak geodesics and subharmonicity of Mabuchi energy. Ann. Fac. Sci. Toulouse Math. (6) 25(5), 935–957 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, X., Păun, M., Zeng, Y.: On deformation of extremal metrics. arXiv:1506.01290

  10. Chen, X., Yuanqi, W.: On the regularity problem of complex Monge–Ampère equations with conical singularities. arXiv:1405.1021

  11. Donaldson, S.: Kähler Metrics with Cone Singularities Along a Divisor. Essays in Mathematics and its Applications, pp. 49–79. Springer, Heidelberg (2012)

    MATH  Google Scholar 

  12. Futaki, A., Honda, S., Saito, S.: Fano-Ricci limit spaces and spectral convergence. arXiv:1509.03862

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

    MATH  Google Scholar 

  14. Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001). Reprint of the 1998 edition

    MATH  Google Scholar 

  15. Hashimoto, Y.: Scalar curvature and Futaki invariant of Kähler metrics with cone singularities along a divisor. arXiv:1508.02640

  16. Keller, J., Zheng, K.: Construction of constant scalar curvature Kähler cone metrics. arxiv:1703.06312

  17. Kobayashi, S.: Transformation Groups in Differential Geometry. Classics in Mathematics. Springer, Berlin (1995). Reprint of the 1972 edition

    MATH  Google Scholar 

  18. Li, L.: Subharmonicity of conic Mabuchi’s functional, I. arXiv:1511.00178

  19. Li, L., Zheng, K.: Generalized Matsushima’s theorem and Kähler–Einstein cone metrics. arXiv:1511.02410

  20. Zheng, K.: Kähler Metrics with Cone Singularities and Uniqueness Problem. Current Trends in Analysis and its Applications, Trends Mathematics. Birkhäuser/Springer, Cham, pp. 395–408 (2015)

Download references

Acknowledgements

We are very grateful to Prof. Xiuxiong Chen, for his continuous encouragement in mathematics, and we would like to thank Prof. S. Donaldson who suggested this problem to us. We also want to thank Dr. Chengjian Yao, Dr. Yu Zeng, and Dr. Yuanqi Wang for lots of useful discussions. L. Li wants to thank Prof. I. Hambleton and Prof. M. Wang for many supports. The work of K. Zheng has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement No. 703949, and was also partially supported by the Engineering and Physical Sciences Research Council (EPSRC) on a Program Grant entitled Singularities of Geometric Partial Differential Equations Reference Number EP/K00865X/1. We would like to thank the anonymous reviewers for their careful reading of our manuscript and constructive comments.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, ON, L8S 4K1, Canada

    Long Li

  2. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

    Kai Zheng

Authors
  1. Long Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kai Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Long Li.

Additional information

Communicated by Ngaiming Mok.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, L., Zheng, K. Uniqueness of constant scalar curvature Kähler metrics with cone singularities. I: reductivity. Math. Ann. 373, 679–718 (2019). https://doi.org/10.1007/s00208-017-1626-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-017-1626-z

Search

Navigation