Convex isoperimetric sets in the Heisenberg group

TL;DR

This paper characterizes convex isoperimetric sets in the Heisenberg group, revealing their boundary structure and regularity properties through advanced geometric and analytical techniques.

Contribution

It provides a novel characterization of convex isoperimetric sets in the Heisenberg group, including boundary foliation by geodesics and Sobolev regularity results.

Findings

Boundaries of convex isoperimetric sets are foliated by geodesics.

Established Sobolev regularity for related vector fields.

Provided a new geometric understanding of isoperimetric sets in the Heisenberg group.

Abstract

We characterize convex isoperimetric sets in the Heisenberg group endowed with horizontal perimeter. We first prove Sobolev regularity for a certain class of vector fields in the plane with bounded variation, related to the curvature equations. Then, by an approximation-reparameterization argument, we show that the boundary of convex isoperimetric sets is foliated by geodesics of the Carnot-Caratheodory distance.