Almost global existence for a fractional Schrodinger equation on spheres and tori

We study the time of existence of the solutions of the following Schr"odinger equation $$i{\psi }_t=(-\mathrm{\Delta }{)}^s\psi +f(\mathrm{\mid }\psi {\mathrm{\mid }}^2)\psi ,x\in {\mathbb{S}}^d,orx\in {\text{T}}^d$$ where $(-\mathrm{\Delta }{)}^s$ stands for the spectrally defined fractional Laplacian with $s>1\mathrm{/}2$ and $f$ a smooth function. We prove an almost global existence result for almost all $s>1\mathrm{/}2$.