Isoperimetric and Sobolev inequalities on hypersurfaces in sub-Riemannian Carnot groups

TL;DR

This paper establishes an isoperimetric inequality for smooth hypersurfaces in Carnot groups, linking geometric properties with horizontal mean curvature, and extends classical inequalities to sub-Riemannian settings.

Contribution

It generalizes classical isoperimetric inequalities to hypersurfaces in Carnot groups, incorporating horizontal mean curvature in a sub-Riemannian context.

Findings

Proved an isoperimetric inequality involving horizontal mean curvature.

Extended classical inequalities to sub-Riemannian Carnot groups.

Discussed applications of the inequality in geometric analysis.

Abstract

Let G be a k-step Carnot group. We prove an isoperimetric-type inequality for compact C^2-smooth immersed hypersurfaces with boundary, involving the horizontal mean curvature of the hypersurface. This generalizes an inequality due to Michael and Simon, and Allard, independently. Some applications are discussed.