Global Regularity of Wave Maps from R to H. Small Energy

We demonstrate that Wave Maps with smooth initial data and small energy from R2+1 to the Lobatchevsky plane stay smooth globally in time. Our method is similar to the one employed in [18]. However, the multilinear estimates required are considerably more involved and present novel technical challenges. In particular, we shall have to work with a modification of the functional analytic framework used in [30], [33], [18]. 1. Formulation of the problem and overview. Let (M, g) be a Riemannian manifold equipped with metric g = (gij). Also, let R, n ≥ 1, be the standard Minkowski space equipped with metric (δij) = diag(−1, 1, . . . , 1). A classical Wave Map u from R to (M, g) is a smooth map which is critical with respect to the functional