  On any compact Riemann surface there always exists a singular non-CSC
(constant scalar curvature) extremal K$\ddot{a}$hler metric which is called a
non-CSC HCMU (the Hessian of the Curvature of the Metric is Umbilical) metric.
In this paper, by moving frames, we show that any non-CSC HCMU metric can not
be isometrically minimal immersed into 3-dimensional real space forms even
locally. In general, any non-CSC HCMU metric can not be isometrically immersed
into 3-dimensional real space forms with constant mean curvature (CMC).
