In this talk, we analyse the exponential stability for the volume preserving mean curvature flow in the flat torus. More precisely, we show that the flow starting near a strictly stable critical set E of the perimeter converges in the long time to a translation of E exponentially fast. We prove this result both for the classical case and for the time-discrete case (Almgren-Taylor-Wang scheme).
An important tool of these proof consists in a new quantitative estimate of Alexandrov type for constant mean curvature hypersurfaces. These works have been done in collaboration with D. De Gennaro, A. Dianaand A. Kubin