Almost Morawetz estimates and global well-posedness for the defocusing $L^2$-critical nonlinear Schr{\"o}dinger equation in higher dimensions

In this paper, we consider the global well-posedness of the defocusing, $L^2$ - critical nonlinear Schr{"o}dinger equation in dimensions $n\ge 3$. Using the I-method, we show the problem is globally well-posed in $n=3$ when $s>2\mathrm{/}5$, and when $n\ge 4$, for $s>\frac{n-2}{n}$. We combine energy increments for the I-method, interaction Morawetz estimates, and almost Morawetz estimates to prove the result.