Isoperimetric Inequalities in Carnot Groups

Abstract

The isoperimetric problem is a very classical problem whose history dates back to more than two thousand years ago. Roughly speaking, the isoperimetric problem is to determine the largest possible area enclosed by a closed curve which has a specified length.

In this thesis, we give proofs of a few theorems on isoperimetric inequalities in Carnot groups. Specifically, for a free nilpotent Carnot group $G$ of step 2, we show the filling function $FA_0(r)$ of the central product $G\cp G$ has a quadratic isoperimetric inequality. Moreover, for Carnot group $G$ of step 2 which satisfies quadratic isoperimetric inequality, we show the filling function of its quotient group satisfies $\fa \preceq r^2\log r$. As a result of two previous theorems, we prove the following result: For a Carnot group $G$ of step $2$, the filling function of the central product $G\times_z G$ satisfies $FA_0 (r) \preceq r^2\log (r)$.