On the Existence of Non-CSC Extremal Kähler Metrics with Finite Singularities on S2

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 α 1 , α 2 , … , α N are N ≥ 4 nonnegative real numbers with α j ≥ 2 ( 1 ≤ j ≤ J ≤ N - 3 ) being integers such that ∑ j = 1 J α j + 2 - N ≥ 0 , given any J points p 1 , … , p J on S 2 \ { 0 , ∞ } , whether there exists a non-CSC conformal HCMU metric g with singular angles 2 π α 1 , … , 2 π α N , which belongs to the first class (see Definition 1.1 ) such that p 1 , … , p J are all saddle points of scalar curvature R of g and 0 , ∞ 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 .