Local well-posedness of the nonlinear Schr\"odinger equations on the sphere for data in modulation spaces

In this paper we discuss a priori estimates derived from the energy method to the initial value problem for the cubic nonlinear Schr"odinger on the sphere $S^2$. Exploring suitable a priori estimates, we prove the existence of solution for data whose regularity is $s=1\mathrm{/}4$.