Area-minimizing horizontal graphs with low-regularity in the sub-Finsler Heisenberg group $\mathbb{H}^1$

TL;DR

This paper constructs examples of area-minimizing horizontal graphs in the sub-Finsler Heisenberg group with low regularity, highlighting the complexity of regularity properties in such geometric contexts.

Contribution

It provides explicit examples of low-regularity, area-minimizing horizontal graphs and cones in the sub-Finsler Heisenberg group, expanding understanding of regularity issues.

Findings

Existence of Lipschitz continuous area-minimizing graphs with singular sets.

Examples of area-minimizing cones in the sub-Finsler setting.

Failure of higher regularity in constructed examples.

Abstract

In the Heisenberg group $\mathbb{H}^1$, equipped with a left-invariant and not necessarily symmetric norm in the horizontal distribution, we provide examples of entire area-minimizing horizontal graphs which are locally Lipschitz in Euclidean sense. A large number of them fail to have further regularity properties. The examples are obtained by prescribing as singular set a horizontal line or a finite union of horizontal half-lines extending from a given point. We also provide examples of families of area-minimizing cones.