Regularity of sets with constant intrinsic normal in a class of Carnot groups

TL;DR

This paper introduces a class of stratified Lie groups called groups of type $ ext{*}$ and proves that sets with constant intrinsic normal are vertical halfspaces, extending rectifiability results to higher-step groups.

Contribution

It defines groups of type $ ext{*}$ and proves that sets with constant intrinsic normal are vertical halfspaces within these groups, generalizing previous results.

Findings

Sets with constant intrinsic normal are vertical halfspaces in groups of type $ ext{*}$

Reduced boundary of finite intrinsic perimeter sets is intrinsically rectifiable

Extends rectifiability results from step 2 to arbitrary step groups

Abstract

In this Note, we define a class of stratified Lie groups of arbitrary step (that are called ``groups of type $\star$'' throughout the paper), and we prove that, in these groups, sets with constant intrinsic normal are vertical halfspaces. As a consequence, the reduced boundary of a set of finite intrinsic perimeter in a group of type $\star$ is rectifiable in the intrinsic sense (De Giorgi's rectifiability theorem). This result extends the previous one proved by Franchi, Serapioni & Serra Cassano in step 2 groups.