The nonlinear Schrödinger equation ground states on product spaces

We study the nature of the nonlinear Schrödinger equation ground states on the product spaces Rn x Mk , where Mk is a compact Riemannian manifold. We prove that for small L2 masses the ground states coincide with the corresponding Rn ground states. We also prove that above a critical mass the ground states have nontrivial Mk dependence. Finally, we address the Cauchy problem issue, which transforms the variational analysis into dynamical stability results.