  In this article, we consider the equivariant Schr\"odinger map from $\Bbb
H^2$ to $\Bbb S^2$ which converges to the north pole of $\Bbb S^2$ at the
origin and spatial infinity of the hyperbolic space. If the energy of the data
is less than $4\pi$, we show that the local existence of Schr\"odinger map.
Furthermore, if the energy of the data sufficiently small, we prove the
solutions are global in time.
