解:根据题意,得$\mathrm {m^2}-4\geqslant 0,$$4 -\mathrm {m^2}\geqslant 0,$即$\mathrm {m^2}\geqslant 4,$$\mathrm {m^2}\leqslant 4,$所以$\mathrm {m^2}=4,$解得$m = \pm 2。$
$ $根据分母不能为$0,$得$m - 2\neq 0,$解得$m\neq 2,$因此$m = - 2。$
$ $此时$n=\frac {0}{-2 - 2}=0,$所以$\sqrt {mn}=\sqrt {-2×0}=0$