Stable soliton resolution for exterior wave maps in all equivariance classes

Abstract

In this paper we consider finite energy ℓ-equivariant wave maps from Rt,x1+3\\(R×B(0,1))→S3 with a Dirichlet boundary condition at r=1, and for all ℓ∈N. Each such ℓ-equivariant wave map has a fixed integer-valued topological degree, and in each degree class there is a unique harmonic map, which minimizes the energy for maps of the same degree. We prove that an arbitrary ℓ-equivariant exterior wave map with finite energy scatters to the unique harmonic map in its degree class, i.e., soliton resolution. This extends the recent results of the first, second, and fourth authors on the 1-equivariant equation to higher equivariance classes, and thus completely resolves a conjecture of Bizoń, Chmaj and Maliborski, who observed this asymptotic behavior numerically. The proof relies crucially on exterior energy estimates for the free radial wave equation in dimension d=2ℓ+3, which are established in the companion paper [13].