In a recent paper, we study the long--time behavior and the stability of the surface diffusion flow of smooth hypersurfaces in the flat torus \(\mathbb T^n\). According to this flow, smooth hypersurfaces move with the outer normal velocity given by the Laplacian of their mean curvature. A first local--in--time existence (and uniqueness) theorem was presented by Escher, Mayer and Simonett, then long--time existence results, in dimensions two and three, were shown by Elliot and Garcke, Wheeler, Acerbi, Fusco and Morini, etc. Even if the three--dimensional case is the most relevant from the physical point of view, since it describes the evolution in time of interfaces between solid phases of a system, driven by the surface diffusion of atoms under the action of a chemical potential, we aim to generalize these results to arbitrary dimensions. More precisely, we show that if the initial set is sufficiently close to a strictly stable critical set for the volume--constrained Area functional and it has ``small energy'', then the flow actually exists for all times and asymptotically converges in a suitable sense to a ``translated'' of the critical set. This is a joint work with Nicola Fusco and Carlo Mantegazza (Scuola Superiore Meridionale \(\&\) Università degli Studi di Napoli Federico II). This seminar is part of the activities of the Excellence Department Project CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.