On the shape of compact hypersurfaces with almost constant mean curvature

TL;DR

This paper establishes a quantitative relationship between the deviation of a hypersurface's mean curvature from constancy and its geometric proximity to tangent balls, aiding the understanding of capillarity problem solutions.

Contribution

It provides a new quantitative stability estimate linking mean curvature oscillation to the hypersurface's shape relative to tangent balls.

Findings

Bound on hypersurface distance from tangent balls based on mean curvature oscillation

Quantitative description of volume-constrained stationary sets in capillarity

Enhanced understanding of hypersurface geometry with almost constant mean curvature

Abstract

The distance of an almost constant mean curvature boundary from a finite family of disjoint tangent balls with equal radii is quantitatively controlled in terms of the oscillation of the scalar mean curvature. This result allows one to quantitatively describe the geometry of volume-constrained stationary sets in capillarity problems.