Scattering of Wave Maps from $\mathbb R^{2+1}$ to general targets

Abstract

We show that smooth, radially symmetric wave maps $U$ from $\mathbb R^{2+1}$ to a compact target manifold $N$, where $\partial_r U$ and $\partial_t U$ have compact support for any fixed time, scatter. The result will follow from the work of Christodoulou and Tahvildar-Zadeh, and Struwe, upon proving that for $\lambda' \in (0,1)$, energy does not concentrate in the set $$K_{5/8T,7/8T}^{\lambda'} = {(x,t) \in \mathbb R^{2+1} | {5pt} |x| \leq \lambda' t, t \in [(5/8)T,(7/8)T]}.$$