Stability of the surface diffusion flow and volume-preserving mean curvature flow in the flat torus

Abstract: We prove that, in the flat torus and in any dimension, the volume-preserving mean curvature flow and the surface diffusion flow, starting $C^{1,1}-$close to a strictly stable critical set of the perimeter $E$, exist for all times and converge to a translate of $E$ exponentially fast as time goes to infinity.