  In this paper we establish the uniformity property of a simplified
Ericksen-Leslie system modelling the hydrodynamics of nematic liquid crystals
on the two dimensional unit sphere $S^2$, namely the uniform convergence in
$L^2$ to a steady state exponentially as t tends to infinity. The main
assumption, similar to Topping [15], concerns the equation of liquid crystal
director $d$ and states that at infinity time, a weak limit $d_\infty$ and any
bubble $\omega_i$($1\le i\le l$) share a common orientation. As consequences,
the uniformity property holds under various types of small initial data.
