# Non-Euclidean isoperimetric inequalities for nilpotent Lie groups

## TL;DR

This paper investigates isoperimetric inequalities in stratified nilpotent Lie groups, revealing dimension-dependent behaviors where Euclidean inequalities hold in lower dimensions but fail in higher ones.

## Contribution

It establishes the existence of a dimension threshold in nilpotent Lie groups where Euclidean isoperimetric inequalities transition from holding to failing.

## Key findings

- Euclidean isoperimetric inequalities hold in all smaller dimensions
- No Euclidean isoperimetric inequality exists in a certain higher dimension
- The results depend on the stratified structure of the Lie groups

## Abstract

This article treats isoperimetric inequalities for integral currents in the setting of stratified nilpotent Lie groups equipped with left-invariant Riemannian metrics. We prove that for each such group there is a dimension in which no Euclidean isoperimetric inequality is admitted, while in all smaller dimensions strictly Euclidean isoperimetric inequalities are satisfied.