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The results of this paper are twofold: In the first part, we prove that forSchr\"odinger map flows from hyperbolic planes to Riemannian surfaces withnon-positive sectional curvatures, the harmonic maps which are holomorphic oranti-holomorphic of arbitrary size are asymptotically stable. In the secondpart, we prove that for Schr\"odinger map flows from hyperbolic planes intoK\"ahler manifolds, the admissible harmonic maps of small size areasymptotically stable. The asymptotic stability results stated here contain twotypes: one is the convergence in $L^{\infty}_x$ as the previous works, theother is convergence to harmonic maps plus radiation terms in the energy space,which is new in literature of Schr\"odinger map flows without symmetryassumptions.

On global dynamics of Schr\"odinger map flows on hyperbolic planes near harmonic maps