Stability of stationary equivariant wave maps from the hyperbolic plane

Abstract

Abstract. In this paper we initiate the study of equivariant wave maps from 2d hyperbolic space, 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 S 2 , we find a family of equivariant harmonic maps H 2 → 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, Qeuc, from R 2 to 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 Qeuc, 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 H 2 , we find a continuous family of asymptotically stable equivariant harmonic maps H 2 → 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.

Convergence to harmonic maps for the Landau-Lifshitz flows between two dimensional hyperbolic spaces

“…In order to state our main theorem, we introduce some notions. [25] is in fact f (z) = λz with λ ∈ (0, 1).…”

Asymptotic stability of harmonic maps between 2D hyperbolic spaces under the wave map equation. II. Small energy case

“…The critical small data Cauchy problem for wave maps on small asymptotically flat perturbations of R 4 to compact Riemann manifolds was studied by Lawrie [30]. The soliton resolution and asymptotic stability of harmonic maps under wave maps on H 2 to S 2 or H 2 in the 1-equivariant case were established by Lawrie, Oh, Shahshahani [31,32,34,35], see also [33] for critical global well-posedness for wave maps from R × H d to compact Riemann manifolds with d ≥ 4.…”

“…Recently Duyckaerts, Jia, Kenig, Merle [11] obtained the universal blow up profile for type II blow up solutions to wave maps u : R × R 2 → S 2 with initial data of energy slightly above the ground state. For wave maps from R × H 2 to H 2 , Lawrie, Oh, Shahshahani [33,34] raised the following soliton resolution conjecture, Conjecture 1.1 Consider the Cauchy problem for wave map u : R × H 2 → H 2 with finite energy initial data (u 0 , u 1 ). Suppose that outside some compact subset K of H 2 for some harmonic map Q : H 2 → H 2 we have u 0 (x) = Q(x), for x ∈ H 2 \K.…”

The Cauchy Problem for Wave Maps on Hyperbolic Space in Dimensions $d\geq 4$

“…In fact, based on the small critical Sobolev data results [36,69,74], a satisfactory large data result on R × R 2 was recently established in [37,64,65,71]. By the same token, we view Theorem 1.1 as a first step towards the interesting case Σ = H 2 , whose study was initiated by the authors [41][42][43] under symmetry assumptions. We refer to Section 1.2 for a more detailed discussion of the history and motivation of the problem.…”