Mathematische Annalen

, Volume 373, Issue 1–2, pp 679–718 | Cite as

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

  • Long LiEmail author
  • Kai Zheng


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.



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.


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Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada
  2. 2.Mathematics InstituteUniversity of WarwickCoventryUK

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