Local well-posedness for nonlinear Schr\"odinger equations on compact product manifolds

We prove new local well-posedness results for nonlinear Schr\"odinger equations posed on a general product of spheres and tori, by the standard approach of multi-linear Strichartz estimates. To prove these estimates, we establish and utilize multi-linear bounds for the joint spectral projector associated to the Laplace--Beltrami operators on the individual sphere factors of the product manifold. To treat the particular case of the cubic NLS on a product of two spheres at critical regularity, we prove a sharp $L^\infty_xL^p_t$ estimate of the solution to the linear Schr\"odinger equation on the two-torus.