We consider 1-equivariant wave maps from R 1+2 → S 2 . For wave maps of topological degree zero we prove global existence and scattering for energies below twice the energy of harmonic map, Q, given by stereographic projection. We deduce this result via the concentration compactness/rigidity method developed by the second author and Merle. In particular, we establish a classification of equivariant wave maps with trajectories that are pre-compact in the energy space up to the scaling symmetry of the equation. Indeed, a wave map of this type can only be either 0 or Q up to a rescaling. This gives a proof in the equivariant case of a refined version of the threshold conjecture adapted to the degree zero theory where the true threshold is 2E(Q), not E(Q). The aforementioned global existence and scattering statement can also be deduced by considering the work of Sterbenz and Tataru in the equivariant setting.For wave maps of topological degree one, we establish a classification of solutions blowing up in finite time with energies less than three times the energy of Q. Under this restriction on the energy, we show that a blow-up solution of degree one is essentially the sum of a rescaled Q plus a remainder term of topological degree zero of energy less than twice the energy of Q. This result reveals the universal character of the known blow-up constructions for degree one, 1-equivariant wave maps of Krieger, the fourth author, and Tataru as well as Raphaël and Rodnianski. η αβ D α ∂ β U = 0, 1991 Mathematics Subject Classification. 35L05, 35L71.

We consider the energy-critical wave maps equation R 1+2 → S 2 in the equivariant case, with equivariance degree k ≥ 2. It is known that initial data of energy < 8πk and topological degree zero leads to global solutions that scatter in both time directions. We consider the threshold case of energy 8πk. We prove that the solution is defined for all time and either scatters in both time directions, or converges to a superposition of two harmonic maps in one time direction and scatters in the other time direction. In the latter case, we describe the asymptotic behavior of the scales of the two harmonic maps.The proof combines the classical concentration-compactness techniques of Kenig-Merle with a modulation analysis of interactions of two harmonic maps in the absence of excess radiation.Theorem 1.2 (Sequential Decomposition). [8,25] Let ψ(t) ∈ H ℓπ be a smooth solution to (1.5) on [0, T + ). Then there exists a sequence of times t n → T + , an integer J ∈ N, a regular map ϕ ∈ H 0 , sequences of scales λ n,j and signs ι j ∈ {−1, 1} for j ∈ {1, . . . , J}, so thatIn the case of finite time blow-up at least one scale λ n,1 → 0 as n → ∞ and ϕ(t) → ϕ(1) is a finite energy map with E( ϕ(1)) = E( ψ) − JE( Q). In the case of a global solution, ϕ(t) can be taken to be a solution to the linear wave equation (2.2) and signs ι j are required to match up so that lim r→∞ ψ(0, r) = ℓπ = lim r→∞ J j=1 ι j Q λn,j (r).Remark 1.3. A decomposition into bubbles for a sequence of times for the full nonequivariant model was obtained by Grinis [21] up to an error that vanishes in a weaker Besov-type norm. Duyckaerts, Jia, Kenig and Merle [12] proved that for energies slightly above E( Q) a one-bubble decomposition holds for continuous time.The same authors obtained in [13] a sequential decomposition into bubbles in the case of the focusing energy critical power-type nonlinear wave equation (NLW).Theorem 1.2 raises two natural questions:• Are there any solutions to (1.5) with J ≥ 2 in (1.7), i.e., are there any solutions that form more than one bubble? • And, if so, does the decomposition (1.7) hold continuously in time, i.e., does soliton resolution hold for (1.5)?

In this paper we establish relaxation of an arbitrary 1-equivariant wave map from R 1+3 t,x \ (R × B(0, 1)) → S 3 of finite energy and with a Dirichlet condition at r = 1, to the unique stationary harmonic map in its degree class. This settles a recent conjecture of Bizoń, Chmaj, Maliborski who observed this asymptotic behavior numerically.1991 Mathematics Subject Classification. 35L05, 35L71.

We consider 1-equivariant wave maps from R 1+2 → S 2 of finite energy. We establish a classification of all degree one global solutions whose energies are less than three times the energy of the harmonic map Q. In particular, for each global energy solution of topological degree one, we show that the solution asymptotically decouples into a rescaled harmonic map plus a radiation term. Together with the companion article, [6], where we consider the case of finite-time blow up, this gives a characterization of all 1-equivariant, degree 1 wave maps in the energy regime [E(Q), 3E(Q)).U (t, r, ω) = (ψ(t, r), ω) → (sin ψ(t, r) cos ω, sin ψ(t, r) sin ω, cos ψ(t, r)), 1991 Mathematics Subject Classification. 35L05, 35L71.

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.