Regularity of sets with constant horizontal normal in the Engel group

TL;DR

This paper proves that sets with constant horizontal normal in the Engel group are rectifiable and can be represented as Lipschitz graphs in certain coordinates, with implications for their geometric and PDE characterizations.

Contribution

It establishes rectifiability of such sets, shows they can be represented as Lipschitz graphs in specific coordinates, and extends results to filiform groups of the first kind.

Findings

Sets with constant horizontal normal are rectifiable in the Engel group.

They can be represented as Lipschitz graphs in specific coordinates.

The rectifiability extends to filiform groups of the first kind.

Abstract

In the Engel group with its Carnot group structure we study subsets of locally finite subRiemannian perimeter and possessing constant subRiemannian normal. We prove the rectifiability of such sets: more precisely we show that, in some specific coordinates, they are upper-graphs of entire Lipschitz functions (with respect to the Euclidean distance). However we find that, when they are written as intrinsic upper-graphs with respect to the direction of the normal, then the function defining the set might even fail to be continuous. Nevertheless, we can prove that one can always find other horizontal directions for which the set is the upper-graph of a function that is Lipschitz-continuous with respect to the intrinsic distance (and in particular H\"older-continuous for the Euclidean distance). We further discuss a PDE characterization of the class of all sets with constant horizontal normal. Finally, we show that our rectifiability argument extends to the case of filiform groups of the first kind.