中文摘要：

HCMU度量是紧黎曼面上带奇点的extremal度量.研究它的存在性十分重要.通过研究Chen和Wu（Pacific J Math,2009,240（2）：267-288）给出的S2上HCMU度量存在的充分必要条件,证明当S2上至少有（N-1）个鞍点时,一定存在non-CSC HCMU度量,其中N是所有锥奇点的个数.

英文摘要：

HCMU metric is an extremal Kahler metric with singularities on a compact Riemann surface. It is important to study the existence of HCMU metrics. Through studying the sufficient and necessary condition of Chen and Wu （ Pacific J Math, 2009,240 （2） ： 267-288 ） for the existence of HCMU metrics on S2, we show that there must exist a non-CSC HCMU metric on S2 which has N conical singularities and at least （ N - 1 ） saddle points.