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

Abstract

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 \(S^{2}\).

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Acknowledgements

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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Correspondence to Zhiqiang Wei.

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Wei, Z., Wu, Y. On the Existence of Non-CSC Extremal Kähler Metrics with Finite Singularities on \(S^2\). J Geom Anal 31, 1555–1567 (2021). https://doi.org/10.1007/s12220-019-00315-y

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Keywords

  • Extremal Kähler metric
  • Conical singularities
  • Cusp singularities

Mathematics Subject Classification

  • 53C21
  • 53C56