On the Existence of Non-CSC Extremal Kähler Metrics with Finite Singularities on \(S^2\)

  • Zhiqiang WeiEmail author
  • Yingyi Wu


We often call an extremal Kähler metric with finite singularities on a compact Riemann surface an HCMU (the Hessian of the Curvature of the Metric is Umbilical) metric. In this paper, we consider the problem: suppose \(\alpha _{1},\alpha _{2},\ldots ,\alpha _{N}\) are \(N\ge 4\) nonnegative real numbers with \(\alpha _{j}\ge 2\;(1\le j\le J\le N-3)\) being integers such that
$$\begin{aligned} \sum _{j=1}^{J}\alpha _{j}+2-N\ge 0, \end{aligned}$$
given any J points \(p_{1},\ldots ,p_{J}\) on \(S^{2}\setminus \{0,\infty \}\), whether there exists a non-CSC conformal HCMU metric g with singular angles\(2\pi \alpha _{1},\ldots ,2\pi \alpha _{N}\), which belongs to the first class (see Definition 1.1) such that \(p_{1},\ldots ,p_{J}\) are all saddle points of scalar curvature R of g and \(0,\infty \) are extremal point of R. We will give a sufficient condition when R has only one saddle point. As its application, we prove that when the number of the singularities is 4, Obstruction Theorem is also a sufficient condition for the existence of a non-CSC conformal HCMU metric on \(S^{2}\).


Extremal Kähler metric Conical singularities Cusp singularities 

Mathematics Subject Classification

53C21 53C56 



Yingyi Wu is supported by NSFC No. 11471308. The authors would like to express their deep gratitude to the referee for his/her very valuable comments on improving the whole paper. This work is also supported by the National Natural Science Foundation of China (Grant No. 11871450).


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Authors and Affiliations

  1. 1.School of Mathematical and StatisticsHenan UniversityKaifengPeople’s Republic of China
  2. 2.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingPeople’s Republic of China

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