Stability of Spherically Symmetric Wave Maps

We study Wave Maps from ${\mathbf{R}}^{2+1}$ to the hyperbolic plane ${\mathbf{H}}^{2}$ with smooth compactly supported initial data which are close to smooth spherically symmetric initial data with respect to some $H^{1+\mu}$, $\mu>0$. We show that such Wave Maps don't develop singularities in finite time and stay close to the Wave Map extending the spherically symmetric data(whose existence is ensured by a theorem of Christodoulou-Tahvildar-Zadeh) with respect to all $H^{1+\delta}, \delta\less\mu_{0}$ for suitable $\mu_{0}(\mu)>0$. We obtain a similar result for Wave Maps whose initial data are close to geodesic ones. This strengthens a theorem of Sideris for this context.