Memoirs of the American Mathematical Society 2006; 80 pp; softcover Volume: 181 ISBN10: 0821838776 ISBN13: 9780821838778 List Price: US$57 Individual Members: US$34.20 Institutional Members: US$45.60 Order Code: MEMO/181/853
 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 ChristodoulouTahvildarZadeh) with respect to all \(H^{1+\delta}, \delta < \mu_{0}\) 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. Table of Contents  Introduction, controlling spherically symmetric wave maps
 Technical preliminaries. Proofs of main theorems
 The proof of Proposition 2.2
 Proof of theorem 2.3
 Bibliography
