Local well-posedness for nonlinear Schrödinger equations on compact product manifolds

We prove new local well-posedness results for nonlinear Schrödinger 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 $L_x^{\mathrm{\infty }}{L}_t^p$ estimate of the solution to the linear Schrödinger equation on the two-torus.