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

Abstract

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

A Heintze--Karcher inequality with free boundaries and applications to capillarity theory

“…More specifically, sets of finite perimeter with distributional mean curvature equal to a constant are shown to be the union of disjoint and tangent balls of equal radius. This result was later quantified for smooth sets in [JN20] with the closeness of the mean curvature to a constant measured in L n (∂Ω), which allowed the authors to show exponential convergence of the volume preserving mean curvature flow to a union of balls in R 3 . For the anisotropic generalization of these results, see [DMMN18,DRKS20].…”