Stability of stationary equivariant wave maps from the hyperbolic plane

In this paper we initiate the study of equivariant wave maps from $2d$ hyperbolic space, ${\Bbb H}^2$, into rotationally symmetric surfaces. This problem exhibits markedly different phenomena than its Euclidean counterpart due to the exponential volume growth of concentric geodesic spheres on the domain. In particular, when the target is ${\Bbb S}^2$, we find a family of equivariant harmonic maps ${\Bbb H}^2\to{\Bbb S}^2$, indexed by a parameter that measures how far the image of each harmonic map wraps around the sphere. These maps have energies taking all values between zero and the energy of the unique co-rotational Euclidean harmonic map, $Q_{{\rm euc}}$, from ${\Bbb R}^2$ to ${\Bbb S}^2$, given by stereographic projection. We prove that the harmonic maps are asymptotically stable for values of the parameter smaller than a threshold that is large enough to allow for maps that wrap more than halfway around the sphere. Indeed, we prove Strichartz estimates for the operator obtained by linearizing around such a harmonic map. However, for harmonic maps with energies approaching the Euclidean energy of $Q_{{\rm euc}}$, asymptotic stability via a perturbative argument based on Strichartz estimates is precluded by the existence of gap eigenvalues in the spectrum of the linearized operator. When the target is ${\Bbb H}^2$, we find a continuous family of asymptotically stable equivariant harmonic maps ${ \Bbb H}^2\to{\Bbb H}^2$ with arbitrarily small and arbitrarily large energies. This stands in sharp contrast to the corresponding problem on Euclidean space, where all finite energy solutions scatter to zero as time tends to infinity.