Long Time Behaviour of the Discrete Volume Preserving Mean Curvature Flow in the Flat Torus

We show that the discrete approximate volume preserving mean curvature flow in the flat torus ${\mathbb{T}}^N$ starting near a strictly stable critical set $E$ of the perimeter converges in the long time to a translate of $E$ exponentially fast. As an intermediate result we establish a new quantitative estimate of Alexandrov type for periodic strictly stable constant mean curvature hypersurfaces. Finally, in the two dimensional case a complete characterization of the long time behaviour of the discrete flow with arbitrary initial sets of finite perimeter is provided.