We study the surface diffusion flow in the flat torus, that is, smooth hypersurfaces moving with the outer normal velocity given by the Laplacian of their mean curvature. This model 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 show that if the initial set is sufficiently ``close'' to a strictly stable critical set for the Area functional under a volume constraint, then the flow actually exists for all times and asymptotically converges to a ``translated'' of the critical set. This generalizes the analogous result in dimension three, by Acerbi, Fusco, Julin and Morini. Joint work with Antonia Diana e Nicola Fusco. This seminar is part of the activities of the Dipartimento di Eccellenza CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.