A rough calculus approach to level sets in the Heisenberg group

TL;DR

This paper develops a new calculus framework for analyzing level sets in the Heisenberg group using rough path inspired equations, enabling measure and area calculations for nonsmooth sets.

Contribution

It introduces novel equations for level sets in the Heisenberg group that extend classical calculus tools to nonsmooth, sub-Riemannian contexts.

Findings

Established well-posedness of the new equations.

Proved an area formula for intrinsic measures.

Derived a coarea formula for level sets.

Abstract

We introduce novel equations, in the spirit of rough path theory, that parametrize level sets of intrinsically regular maps on the Heisenberg group with values in $\mathbb{R}^2$. These equations can be seen as a sub-Riemannian counterpart to classical ODEs arising from the implicit function theorem. We show that they enjoy all the natural well-posedness properties, thus allowing for a "good calculus" on nonsmooth level sets. We apply these results to prove an area formula for the intrinsic measure of level sets, along with the corresponding coarea formula.