  We consider the Cauchy problem for the Schr\"odinger maps evolution when the
domain is the hyperbolic plane. An interesting feature of this problem compared
to the more widely studied case on the Euclidean plane is the existence of a
rich new family of finite energy harmonic maps. These are stationary solutions,
and thus play an important role in the dynamics of Schr\"odinger maps. The main
result of this article is the asymptotic stability of (some of) such harmonic
maps under the Schr\"odinger maps evolution. More precisely, we prove the
nonlinear asymptotic stability of a finite energy equivariant harmonic map $Q$
under the Schr\"odinger maps evolution with respect to non-equivariant
perturbations, provided $Q$ obeys a suitable linearized stability condition.
This condition is known to hold for all equivariant harmonic maps with values
in the hyperbolic plane and for a subset of those maps taking values in the
sphere. One of the main technical ingredients in the paper is a global-in-time
local smoothing and Strichartz estimate for the operator obtained by
linearization around a harmonic map, proved in the companion paper [36].
