Isoperimetric stability
We obtain here a generalization of Bonnessen's inequality; this inequality says that if \(D\) is a convex domain of the plane then there exists a circular annulus of width \(d\) which contains \(\partial D\) such that \[ {(\text{Length }\partial D)^2\over 4\pi}-\text{Area }D\geq d^2. \] A compact smooth domain of a Riemannian manifold is called geometrically stable if \(\partial D\) has constant mean curvature, and if it is stable with a nullity space for the second variation the area is exactly described by the isometric variation of \(D\). We prove a Bonnessen inequality for domains in a \(C^1\)-neighbourhood of a geometrically stable domain. For example, we obtain that geodesic balls of the complex hyperbolic space realize locally the isoperimetric inequality.
scientific article; zbMATH DE number 952633