On the shape of compact hypersurfaces 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.

36 pages, 2 figures. In this version we have added an appendix about almost umbilical surfaces