Sub-Finsler Heisenberg Perimeter Measures

TL;DR

This paper investigates perimeter measures in the Heisenberg group with sub-Finsler metrics, reducing Minkowski content to Lebesgue surface integrals, and provides evidence supporting Pansu's conjecture in the sub-Finsler setting.

Contribution

It introduces a reduction of Minkowski content to Lebesgue surface integrals and explores perimeter measures in sub-Finsler metrics, extending Pansu's conjecture to this broader context.

Findings

Minkowski content can be expressed as an integral over Lebesgue surface area.

Existence of surfaces with Legendrian foliation generalizing Pansu's bubble set.

Examples show lower isoperimetric ratios than Pansu's bubble set in sub-Finsler metrics.

Abstract

This work is an investigation of perimeter measures in the metric measure space given by the Heisenberg group with Haar measure and a Carnot-Carath\'eodory metric, which is in general a sub-Finsler metric. Included is a reduction of Minkowski content in any CC-metric to an integral formula in terms of Lebesgue surface area in $\mathbb{R}^3$. Using this result, I study two perimeter measures that arise from the study of Finsler normed planes, and provide evidence that Pansu's conjecture regarding the isoperimetric problem in the sub-Riemannian case appears to hold in the more general sub-Finsler case. This is contrary to the relationship between Finsler and Riemannian isoperimetrices. In particular, I show that for any CC-metric there exist a class of surfaces with Legendrian foliation by CC-geodesics generalizing Pansu's bubble set, but that even in their natural metric using either perimeter measure, the computed examples of such surfaces have lower isoperimetric ratio than the Pansu bubble set, which has Legendrian foliation by sub-Riemannian geodesics.