Monotone sets and local minimizers for the perimeter in Carnot groups

TL;DR

This paper extends the study of monotone sets from the Heisenberg group to general Carnot groups, showing they are perimeter-minimizing sets with specific measure-theoretic and topological properties.

Contribution

It characterizes monotone sets in Carnot groups as perimeter local minimizers and explores their measure-theoretic and topological features, generalizing previous results.

Findings

Monotone sets are sets with locally finite perimeter in Carnot groups.

Under certain conditions, monotone sets have measure-theoretic interiors and supports that are exactly monotone.

The paper provides conditions for monotone sets to have measure-theoretic representatives that are precisely monotone.

Abstract

Monotone sets have been introduced about ten years ago by Cheeger and Kleiner who reduced the proof of the non biLipschitz embeddability of the Heisenberg group into $L^1$ to the classification of its monotone subsets. Later on, monotone sets played an important role in several works related to geometric measure theory issues in the Heisenberg setting. In this paper, we work in an arbitrary Carnot group and show that its monotone subsets are sets with locally finite perimeter that are local minimizers for the perimeter. Under an additional condition on the ambient Carnot group, we prove that their measure-theoretic interior and support are precisely monotone. We also prove topological and measure-theoretic properties of local minimizers for the perimeter whose interest is independent from the study of monotone sets. As a combination of our results, we get in particular a sufficient condition under which any monotone set admits measure-theoretic representatives that are precisely monotone.