Rectifiability of sets of finite perimeter in Carnot groups: existence of a tangent hyperplane

TL;DR

This paper proves that in Carnot groups, sets of finite perimeter have tangent hyperplanes at almost every point, extending previous results in step 2 Carnot groups to more general cases.

Contribution

It extends the existence of tangent hyperplanes for sets of finite perimeter from step 2 Carnot groups to general Carnot groups.

Findings

Almost every point has a tangent that is a vertical halfspace.

Extension of previous step 2 results to general Carnot groups.

Provides a geometric understanding of perimeter sets in Carnot groups.

Abstract

We consider sets of locally finite perimeter in Carnot groups. We show that if E is a set of locally finite perimeter in a Carnot group G, then for almost every x in G with respect to the perimeter measure of E, some tangent of E at x is a vertical halfspace. This is a partial extension of a theorem of Franchi-Serapioni-Serra Cassano in step 2 Carnot groups: they have shown that, for almost every x, E has a unique tangent at x, and this tangent is a vertical halfspace.