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

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