Roughness of level sets of differentiable maps on Heisenberg group

TL;DR

This paper studies the metric and measure-theoretic properties of level sets of horizontally differentiable maps on the Heisenberg group, revealing their potential irregularity and measure-zero or infinite measure characteristics.

Contribution

It provides a detailed analysis of level sets of maps with surjective horizontal differential on the Heisenberg group, highlighting their rough geometric nature.

Findings

Level sets are locally simple curves with Hausdorff dimension 2.

Two-dimensional Hausdorff measure of level sets can be zero or infinite.

Level sets may not be intrinsic regular manifolds.

Abstract

We investigate metric properties of level sets of horizontally differentiable maps defined on the first Heisenberg group $(\Bbb{H}^1,d_{cc})$ equipped with the standard sub-Riemannian structure. In particular, we present an exhaustive analysis in a new case of a map $F\in C^1_H(\Bbb{H}^1, \Bbb{R}^2)$ with surjective horizontal differential (an analogue of the classical implicit function theorem). Among other results, we show that a level set of such map is locally a simple curve of Hausdorff sub-Riemannian dimension 2, but, surprisingly, in general its two-dimensional Hausdorff measure can be zero or infinity. Therefore, those level sets (called \textsf{vertical curves}) can be of rough nature and not belong to the class of intrinsic regular manifolds.