Dimension comparison and H-regular surfaces in Heisenberg groups

Abstract

In this thesis we study a specific Carnot group which is the $n$-th Heisenberg group $\mathbb{H}^n = (\mathbb{R}^{2n+1}, \ast)$. Carnot groups are simply connected nilpotent Lie groups whose Lie algebra admits a stratification. The Heisenberg group $\mathbb{H}^n$ is one of the easiest examples of non-commutative Carnot groups.

In the first part of the thesis we recall some preliminaries about measure theory and Heisenberg groups which we need. We also prove some useful inequalities which are needed to prove the main theorem.

The main result of the thesis is that in any Heisenberg group $\mathbb{H}^n$ there exists a $\mathbb{H}$-regular hypersurface which has a Euclidean Hausdorff dimension of $2n + \frac{1}{2}$. This generalizes a construction in [KSC04] from $n = 1$ to $n > 1$. To prove the main result we need a dimension comparison theorem in general Heisenberg groups. We will prove such a dimension comparison theorem in the thesis combining ideas from the proofs in [BRSCO03](for $\mathbb{H}^1$) and [BTW09](for Carnot groups).

Dimension comparison theorem gives us information about the absolute continuity of the Hausdorff measure when comparing the measure in Euclidean and Heisenberg point of view. As a corollary we obtain Hausdorff dimension comparison which gives us lower and upper bounds for the Heisenberg Hausdorff dimension of a set $A \subset \mathbb{H}^n$. More precisely the results are about comparing Hausdorff measures and dimension when computed in the Euclidean and the Heisenberg distance, respectively.