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

TL;DR

This paper establishes asymptotic stability of certain harmonic maps under Schr"odinger map flows from hyperbolic planes to various target surfaces, including Riemannian and K"ahler manifolds, with new convergence results in energy space.

Contribution

It proves the asymptotic stability of harmonic maps under Schr"odinger flows from hyperbolic planes to Riemannian and K"ahler targets, including convergence to harmonic maps plus radiation, without symmetry assumptions.

Findings

Holomorphic and anti-holomorphic harmonic maps are asymptotically stable.

Small harmonic maps into K"ahler manifolds are asymptotically stable.

New convergence to harmonic maps plus radiation in energy space.

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.