Quantitative Alexandrov theorem and asymptotic behavior of the volume preserving mean curvature flow

We prove a new quantitative version of the Alexandrov theorem which states that if mean curvature regular set in R^{n+1} is close to constant L^{n}-sense, then union disjoint balls with respect Hausdorff distance. This result more general than previous quantifications and using it we are able show R^2 R^3 weak solution volume preserving flow starting from finite perimeter asymptotically convergences equisize balls, up possible translations. Here by flat flow, obtained via minimizing movements scheme.