HCMU是一种在Riemann面上带奇点的extremal度量．在面积和Calabi能量有界的情况下， HCMU的Gauss曲率是Riemann面上的连续函数．本文得到一个在球面上没有Gauss曲率鞍点的HCMU的明显表达式，并进一步证明了在球面或环面上HCMU的Gauss曲率光滑的充要条件是度量的所有奇点的角度都是整数．

An HCMU is a kind of extremal metric with singularities on a Riemann surface. If the area and Calabi energy are both bounded, the Gauss curvature of an HCMU is a continuous function on the Riemann surface. In this paper we get an explicit construction of an HCMU on S² which has no saddle point of the Gauss curvature of the metric. Further more we prove that on S² or T² the Gauss curvature of an HCMU is smooth if and only all of the singular angles are integers.