Contact equations, Lipschitz extensions and isoperimetric inequalities

TL;DR

This paper characterizes Lipschitz mappings via PDEs, explores their extensions in graded groups, and connects these to isoperimetric inequalities, revealing new geometric and analytic insights.

Contribution

It introduces a PDE-based framework for Lipschitz extensions in graded groups and establishes quadratic isoperimetric inequalities in Allcock groups.

Findings

Lipschitz mappings characterized by nonlinear PDEs

Existence of Lipschitz extensions in certain 2-step groups

Quadratic isoperimetric inequalities in Allcock groups

Abstract

We characterize locally Lipschitz mappings and existence of Lipschitz extensions through a first order nonlinear system of PDEs. We extend this study to graded group-valued Lipschitz mappings defined on compact Riemannian manifolds. Through a simple application, we emphasize the connection between these PDEs and the Rumin complex. We introduce a class of 2-step groups, satisfying some abstract geometric conditions and we show that Lipschitz mappings taking values in these groups and defined on subsets of the plane admit Lipschitz extensions. We present several examples of these groups, called Allcock groups, observing that their horizontal distribution may have any codimesion. Finally, we show how these Lipschitz extensions theorems lead us to quadratic isoperimetric inequalities in all Allcock groups.