Global well-posedness for rough solutions of defocusing cubic NLS on three-dimensional compact manifolds

In this paper, we investigate the global well-posedness for cubic nonlinear Schrödinger (NLS) equation [Formula: see text] posed on the three-dimensional compact manifolds [Formula: see text] with initial data [Formula: see text] where [Formula: see text] for Zoll manifold and [Formula: see text] for the product of spheres [Formula: see text]. We utilize the multilinear eigenfunction estimate on compact manifold to treat the interaction of different frequencies, which is more complicated compared to the case of flat torus [C. Fan, G. Staffilani, H. Wang and B. Wilson, On a bilinear Strichartz estimate on irrational tori, Anal. PDE 11 (2018) 919–944] and waveguide manifold [Z. Zhao and J. Zheng, Long time dynamics for defocusing cubic nonlinear Schrödinger equations on three dimensional product space, SIAM J. Math. Anal. 53 (2020) 3644–3660]. Moreover, combining with the I-method adapted to the non-periodic case, bilinear Strichartz estimates along with the scale-invariant [Formula: see text] linear Strichartz estimates, we partially obtain the similar result of [Z. Zhao and J. Zheng, Long time dynamics for defocusing cubic nonlinear Schrödinger equations on three dimensional product space, SIAM J. Math. Anal. 53 (2020) 3644–3660] on non-flat compact manifold setting. As a consequence, we obtain the polynomial bounds of the [Formula: see text] norm of solution [Formula: see text].