  The results of this paper are twofold: In the first part, we prove that for
Schr\"odinger map flows from hyperbolic planes to Riemannian surfaces with
non-positive sectional curvatures, the harmonic maps which are holomorphic or
anti-holomorphic of arbitrary size are asymptotically stable. In the second
part, we prove that for Schr\"odinger map flows from hyperbolic planes into
K\"ahler manifolds, the admissible harmonic maps of small size are
asymptotically stable. The asymptotic stability results stated here contain two
types: one is the convergence in $L^{\infty}_x$ as the previous works, the
other is convergence to harmonic maps plus radiation terms in the energy space,
which is new in literature of Schr\"odinger map flows without symmetry
assumptions.
