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

Abstract

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.