On scattering for the defocusing nonlinear Schr\"odinger equation on waveguide $ \mathbb{R}^m$ $\times$ $\mathbb{T}$ (when $m=2,3$)

In the article, we prove the large data scattering for two problems, i.e. the defocusing quintic nonlinear Schr{\"o}dinger equation on $\mathbb{R}^2$ $\times$ $\mathbb{T}$ and the defocusing cubic nonlinear Schr{\"o}dinger equation on $\mathbb{R}^3$ $\times$ $\mathbb{T}$. Both of the two equations are mass supercritical and energy critical. The main ingredients of the proofs contain global Stricharz estimate, profile decomposition and energy induction method. This paper is the second project of our series work (two papers, together with [36]) on large data scattering for the defocusing critical NLS with integer index nonlinearity on low dimensional waveguides. At this point, that type of problems are almost solved except for two remaining resonant system conjectures and the quintic NLS problem on $\mathbb{R}\times \mathbb{T}$.