Uniqueness of martingale solutions for the stochastic nonlinear Schr\"odinger equation on 3d compact manifolds

We prove pathwise uniqueness for solutions of the nonlinear Schr"{o}dinger equation with conservative multiplicative noise on compact 3D manifolds. In particular, we generalize the result by Burq, G'erard and Tzvetkov (N. Burq, P. G'erard, and N. Tzvetkov. Strichartz inequalities and the nonlinear Schr"{o}dinger equation on compact manifolds. American Journal of Mathematics, 126 (3):569–605, 2004) to the stochastic setting. The proof is based on deterministic and stochastic Strichartz estimates and the Littlewood-Paley decomposition.