A sharp quantitative Alexandrov inequality and applications to volume preserving geometric flows in 3D

TL;DR

This paper establishes a sharp quantitative Alexandrov inequality in 3D and applies it to analyze the asymptotic behavior of volume-preserving geometric flows like mean curvature and Mullins-Sekerka flows.

Contribution

The paper introduces a new sharp quantitative Alexandrov inequality in three dimensions and uses it to study the asymptotic behavior of specific volume-preserving geometric flows.

Findings

Established a 3D sharp quantitative Alexandrov inequality.

Analyzed asymptotic behavior of volume-preserving mean curvature flow.

Analyzed asymptotic behavior of Mullins-Sekerka flat flow.

Abstract

We study the asymptotic behavior of the volume preserving mean curvature and the Mullins-Sekerka flat flow in three dimensional space. Motivated by this we establish a 3D sharp quantitative version of the Alexandrov inequality for $C^2$-regular sets with a perimeter bound.