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 Schr\"{o}dinger equation(NLS) i∂ₜu+Δgu=|u|²u posed on the three dimensional compact manifolds $(M,g)$ with initial data u₀∈ Hˢ(M) where s>√21-14 for Zoll manifold and s>1+3√58 for the product of spheres S²×S¹. 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, B. Wilson, Anal. PDE, 11 (2018), 919-944.] and waveguide manifold [Z. Zhao, J. Zheng, 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 Lᵖ linear Strichartz estimates, we partially obtain the similar result of [Z. Zhao, J. Zheng, SIAM J. Math. Anal. 53 (2020), 3644-3660.] on non-flat compact manifold setting. As a consequence, we obtain the polynomial bounds of the Hˢ norm of solution u.