Vertical curves and vertical fibers in the Heisenberg group

TL;DR

This paper investigates the geometric and measure-theoretic properties of vertical curves and fibers in the Heisenberg group, revealing their local structure, dimension variability, and contrasting behavior with Euclidean projections.

Contribution

It characterizes vertical curves via cone conditions, relates them to intrinsic Lipschitz graphs, and demonstrates their Hausdorff dimension variability, along with analyzing fiber measures of certain maps.

Findings

Connected vertical curves are locally biHölder to intervals.

Vertical curves can have Hausdorff dimension different from 2.

Existence of maps with fibers of arbitrarily small measure close to projections.

Abstract

Let $\mathbb H$ denote the three-dimensional Heisenberg group. In this paper, we study vertical curves in $\mathbb H$ and fibers of maps $\mathbb H \to \mathbb R^2$ from a metric perspective. We say that a set in $\mathbb H$ is a vertical curve if it satisfies a cone condition with respect to a homogeneous cone with axis $\langle Z \rangle$, the center of $\mathbb H$. This is analogous to the cone condition used to define intrinsic Lipschitz graphs. In the first part of the paper, we prove that connected vertical curves are locally biH\"older equivalent to intervals. We also show that the class of vertical curves coincides with the class of intersections of intrinsic Lipschitz graphs satisfying a transversality condition. Unlike intrinsic Lipschitz graphs, the Hausdorff dimension of a vertical curve can vary; we construct vertical curves with Hausdorff dimension either strictly larger or strictly smaller than 2. Consequently, there are intersections of intrinsic Lipschitz graphs with Hausdorff dimension either strictly larger or strictly smaller than 2. In the second part of the paper, we consider smooth functions $\beta$ from the unit ball $B$ in $\mathbb H$ to $\mathbb R^2$. We show that, in contrast to the situation in Euclidean space, there are maps such that $\beta$ is arbitrarily close to the projection $\pi$ from $\mathbb H$ to the horizontal plane, but the average $\mathcal{H}^2$ measure of a fiber of $\beta$ in $B$ is arbitrarily small.