Nonlinear Schrödinger Equations on Compact Manifolds

Abstract

Nonlinear Schrödinger equations have been studied by mathematicians for about thirty years. However, until recently, most of the contributions concerned the equation on the whole Euclidean space, with the notable exception of J. Bourgain’s contributions on tori. In the case of general Riemannian manifolds, the interaction of geometry with nonlinear operations leads to new phenomena, particularly if the manifold is compact. Here we review the state of the art concerning the Cauchy problem on such manifolds, and we describe optimal results on spheres, where new estimates on spherical harmonics play a crucial role. The matter of this paper is based on a series of results in collaboration with N. Burq and N. Tzvetkov.