Global Regularity of Wave Maps from R³⁺¹ to H²

We consider Wave Maps with smooth compactly supported initial data of small H^{{3/2}}-norm from R^{3+1} to the hyperbolic plane and show that they stay smooth globally in time. Our methods are based on the introduction of a global Coulomb Gauge as in a recent paper by Shatah and Struwe, followed by dynamic separation as in earlier work of Klainerman and Machedon. We then rely on an adaptation of T.Tao's methods used in his recent breakthrough result on Wave Maps to spheres in low spatial dimensions.