Geometric inequalities in Carnot groups

TL;DR

This paper establishes various geometric inequalities for smooth hypersurfaces in Carnot groups, a class of sub-Riemannian manifolds, using the $ ext{ extonehalf}$-perimeter measure, advancing understanding of geometric analysis in these structures.

Contribution

It introduces new geometric inequalities for hypersurfaces in Carnot groups, expanding the theoretical framework of sub-Riemannian geometry.

Findings

Proved several inequalities involving hypersurfaces in Carnot groups.

Extended geometric analysis tools to sub-Riemannian settings.

Enhanced understanding of perimeter measures in Carnot groups.

Abstract

Let $\GG$ be a sub-Riemannian $k$-step Carnot group of homogeneous dimension $Q$. In this paper, we shall prove several geometric inequalities concerning smooth hypersurfaces (i.e. codimension one submanifolds) immersed in $\GG$, endowed with the $\HH$-perimeter measure.