Isoperimetric inequalities in cylinders with density

Katherine Castro

doi:10.1016/j.na.2021.112726

Abstract

Given a compact Riemannian manifold with density M without boundary and the real line R with constant density, we prove that isoperimetric regions of large volume in M×R with the product density are slabs of the form M×[a,b]. We previously prove, as a necessary step, the existence of isoperimetric regions in any manifold of density where a subgroup of the group of transformations preserving weighted perimeter and volume acts cocompactly.

Highlights

Introduction and preliminariesIn recent years, isoperimetric problems have been considered in manifolds with density
In 1997 Bobkov [4] proved a functional version of this isoperimetric inequality, later extended to the sphere and used to prove isoperimetric estimates for the unit cube by Barthe and Maurey [2]
Bakry and Ledoux [1] and Bayle [3] proved generalizations of the Levy–Gromov isoperimetric inequality and other geometric comparisons depending on a lower bound on the generalized Ricci curvature of the manifold

Summary

Introduction and preliminaries

Isoperimetric problems have been considered in manifolds with density. The proof of existence is based on Galli and Ritore’s in contact sub-Riemannian manifolds, see [12] We prove the existence of isoperimetric sets, for any volume, in a manifold with density (M, g, Ψ) such that Isom(M, g, Ψ) acts cocompactly. In a manifold with density (M, g, Ψ) such that Isom(M, g, Ψ) acts cocompactly, isoperimetric sets exist for any given volume.

Note that

Hence we have