A NOTE ON ILL-POSEDNESS OF THE CAUCHY PROBLEM FOR HEISENBERG WAVE MAPS

Abstract. We introduce a notion of wave maps with target in the sub-Riemannian Heisenberg group and study their relation with Riemannian wave maps with range in Lagrangian submanifolds. As an application we establish existence and eventually ill-posedness of the corresponding Cauchy problem. 1. Introduction. In the last two decades there have been many contributions to the study of wave maps (see [15] and references therein). In general, well posedness for the Cauchy problem holds under the condition that the target manifold has bounded geometry. In the present paper we study a simple, explicit model of singular target space