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 $S^{2}$.