The Cauchy Problem for the Nonlinear Schrödinger Equation on a Compact Manifold

We discuss the wellposedness theory of the Cauchy problem for the nonlinear Schro- dinger equation on compact Riemannian manifolds. New dispersive estimates on the linear Schrodinger group are used to get global existence in the energy space on arbit- rary surfaces and three-dimensional manifolds, generalizing earlier results by Bourgain on tori. On the other hand, on specific manifolds such as spheres, new instability phenomena are displayed, leading to some kind of illposednesss in higher dimensions.