Isoperimetric inequalities in cylinders with density

TL;DR

This paper proves that in certain weighted product spaces, large-volume isoperimetric regions are slabs, extending understanding of geometric inequalities in manifolds with density and establishing existence results under symmetry conditions.

Contribution

It establishes that large-volume isoperimetric regions in manifolds with density are slabs and proves existence of such regions under symmetry assumptions.

Findings

Large-volume isoperimetric regions are slabs in manifolds with density.

Existence of isoperimetric regions is proven under cocompact symmetry conditions.

Results extend isoperimetric inequalities to weighted product manifolds.

Abstract

Given a compact Riemannian manifold with density $M$ without boundary and the real line $\mathbb{R}$ with constant density, we prove that isoperimetric regions of large volume in $M\times\mathbb{R}$ with the product density are slabs of the form $M\times [a,b]$. 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.