Volume preserving mean curvature flows near strictly stable sets in flat torus

In this paper we establish a new stability result for the smooth volume preserving mean curvature flow in flat torus ${\mathbb{T}}^n$ in low dimensions $n=3,4$. The result says roughly that if the initial set is near to a strictly stable set in ${\mathbb{T}}^n$ in $H^3$-sense, then the corresponding flow has infinite lifetime and converges exponentially fast to a translate of the strictly stable (critical) set in $W^{2,5}$-sense.