Stability of Spherically Symmetric Wave Maps

We study Wave Maps from R2+1 to the hyperbolic plane H2 with smooth compactly supported initial data which are close to smooth spherically symmetric initial data with respect to some H1+μ, μ > 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 H1+δ , δ < μ0 for suitable μ0(μ) > 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.