Asymptotic Stability of Harmonic Maps on the Hyperbolic Plane under the Schrödinger Maps Evolution

Abstract

We consider the Cauchy problem for the Schrödinger 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 Schrödinger maps. The main result of this article is the asymptotic stability of (some of) such harmonic maps under the Schrödinger maps evolution. More precisely, we prove the nonlinear asymptotic stability of a finite energy equivariant harmonic map Q under the Schrödinger 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].

On global dynamics of Schrödinger map flows on hyperbolic planes near harmonic maps

“…We see the topology of perturbations assumed here is almost the same as that used in [26] in the angle direction and stronger in the radial direction. Moreover, [Remark 1.8, [26]] pointed out that non-equivariant harmonic maps Q seem to require new ideas. Theorem 1.1 here covers all holomorphic maps and anti-holomorphic maps of compact images, which contain wide class of non-equivariant harmonic maps.…”

“…Remark 1.1. Results as (1.3) were first established by Tao [57] for the wave map equation on R 2 , and later obtained by [26,[30][31][32]36] in the setting of wave maps/ Schrödinger map flows on hyperbolic spaces. The type result (1.4) is new in the setting of non-equivariant Schrödinger map flows on both Euclidean spaces and curved base manifolds.…”

“…And the first non-equivariant data result on hyperbolic spaces was obtained by Lawrie-Oh-Shahshahani [30] where optimal small data global theory for high dimensional wave maps on hyperbolic spaces was established. More recently, Lawrie-Luhrmann-Oh-Shahshahani [25,26] studied global dynamics of SL under the nonequivariant perturbations of strongly linear stable equivariant harmonic maps from H 2 to rationally symmetric Riemannian surfaces. Their works exploit the delicate smoothing effects of Schrödinger equations.…”

“…Holomorphic maps and anti-holomorphic maps are typical harmonic maps between Kähler manifolds. A remarkable observation of [26] reveals that linearized operators around holomorphic maps between Riemannian surfaces are self-adjoint. We focus on holomorphic maps and anti-holomorphic maps to take this convenience for large data.…”

“…In the work [26], for equivariant harmonic maps Q (which are especially holomorphic) and rotationally symmetric Riemannian surfaces N , the stability was proved for non-equivariant initial data u 0 satisfying u 0 − Q H 1+δ + D Ω u 0 H 1+2δ ≪ 1, where D Ω can be viewed as a derivative measuring how co-rotational u 0 is. We see the topology of perturbations assumed here is almost the same as that used in [26] in the angle direction and stronger in the radial direction. Moreover, [Remark 1.8, [26]] pointed out that non-equivariant harmonic maps Q seem to require new ideas.…”