Isoperimetric Regions in Anisotropically Scaled Product Manifolds

TL;DR

This paper investigates the shape of minimal boundary regions in scaled product manifolds, showing that under certain conditions, these regions are products of the original manifolds and their isoperimetric regions.

Contribution

It establishes that in small homothetic products of manifolds, isoperimetric regions are products of the factors and their isoperimetric regions, assuming smooth boundaries.

Findings

Isoperimetric regions in scaled product manifolds are products of the factors.

Results depend on the size of the homothetic copy and boundary smoothness.

Provides conditions under which the product structure of isoperimetric regions holds.

Abstract

Let $M, N$ be compact Riemannian manifolds. Then, for fixed volume fraction, in the product of a sufficiently small homothetic copy of $M$ with $N$, every isoperimetric region is the product of $M$ with an isoperimetric region in $N$, provided the boundaries of the isoperimetric regions in $N$ are smooth.