Isoperimetric stability
type: Item
instance of: scholarly article
title: Isoperimetric stability
zbMATH Open document ID: 0865.53027
DOI: 10.1007/BF01445253
author: Gilles Carron
published in: Mathematische Annalen
publication date: 1997-07-10T00:00:00Z
full work available at URL: https://eudml.org/doc/165456
review text: 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.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C65
zbMATH DE Number: 952633
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: Bonnessen inequality
zbMATH Keywords: geometrically stable domain
zbMATH Keywords: geodesic balls
zbMATH Keywords: complex hyperbolic space
zbMATH Keywords: isoperimetric inequality
MaRDI profile type: Publication
cites work: Stability of hypersurfaces of constant mean curvature in Riemannian manifolds
cites work: An inequality of ``Reilly''-type for submanifolds of the hyperbolic space
cites work: Stability in the Isoperimetric Problem for Convex or Nearly Spherical Domains in R n
cites work: Bonnesen's inequality for the isoperimetric deficiency of closed curves in the plane
cites work: Riemannian geometry
cites work: Certain conditions for a Riemannian manifold to be isometric with a sphere
cites work: A strong form of the isoperimetric inequality in Rn
cites work: Q5548503
cites work: Elastic waves in rods and clad rods
OpenAlex ID: W4252623686
description: scientific article; zbMATH DE number 952633