The NLS ground states on product spaces

We study the nature of the Nonlinear Schr"odinger equation ground states on the product spaces ${\mathbb{R}}^n\times M^k$, where $M^k$ is a compact Riemannian manifold. We prove that for small $L^2$ masses the ground states coincide with the corresponding ${\mathbb{R}}^n$ ground states. We also prove that above a critical mass the ground states have nontrivial $M^k$ dependence. Finally, we address the Cauchy problem issue which transform the variational analysis to dynamical stability results.