  In this article, we investigate the global well-posedness for cubic nonlinear
Schr\"{o}dinger equation(NLS) $ i\partial_tu+\Delta_gu=|u|^2u$ posed on the
three dimensional compact manifolds $(M,g)$ with initial data $u_0\in H^s(M)$
where $s>\frac{\sqrt{21}-1}{4}$ for Zoll manifold and $s>\frac{1+3\sqrt{5}}{8}$
for the product of spheres $\Bbb{S}^2\times\Bbb{S}^1$. 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
$L^p$ 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 $H^s$ norm of solution $u$.
