  We study the global existence and stability of surface diffusion flow (the
normal velocity is given by the Laplacian of the mean curvature) of smooth
boundaries of subsets of the $n$--dimensional flat torus. More precisely, we
show that if a smooth set is ``close enough'' to a strictly stable critical set
for the Area functional under a volume constraint, then the surface diffusion
flow of its boundary hypersurface exists for all time and asymptotically
converges to the boundary of a ``translated'' of the critical set. This result
was obtained in dimension $n=3$ by Acerbi, Fusco, Julin and Morini (extending
previous results for spheres of Escher, Mayer and Simonett and Elliott and
Garcke in dimension $n=2$). Our work generalizes such conclusion to any
dimension $n\in\mathbb N$. For sake of clarity, we show all the details in
dimension $n=4$ and we list the necessary modifications to the quantities
involved in the proof in the general $n$--dimensional case, in the last
section.
