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

TL;DR

This paper establishes a new quantitative version of the Alexandrov theorem relating mean curvature closeness to geometric proximity to unions of balls, and analyzes the asymptotic behavior of volume-preserving mean curvature flow in low dimensions.

Contribution

It introduces a more general quantitative Alexandrov theorem and demonstrates convergence of weak solutions of volume-preserving mean curvature flow to unions of equal-sized balls in R^2 and R^3.

Findings

Quantitative Alexandrov theorem for sets with mean curvature close to constant.

Weak solutions of volume-preserving mean curvature flow converge to unions of equal-sized balls.

Results extend previous quantifications and apply to low-dimensional cases.

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.