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

In this article, we investigate the global well-posedness for cubic nonlinear Schrödinger equation(NLS) [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 [23] and waveguide manifold [47]. 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 [47] 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].