Generalized mean curvature flow in Carnot groups

TL;DR

This paper extends the theory of mean curvature flow to Carnot groups within sub-Riemannian geometry, adapting Euclidean methods to establish fundamental properties of the flow.

Contribution

It introduces a generalized mean curvature flow framework in Carnot groups, including level set and viscosity solutions, with key comparison principles and existence results.

Findings

Established comparison principles for the flow

Proved existence and uniqueness of solutions

Analyzed basic geometric properties of the flow

Abstract

In this paper we study the generalized mean curvature flow of sets in the sub-Riemannian geometry of Carnot groups. We extend to our context the level sets method and the weak (viscosity) solutions introduced in the Euclidean setting by Evans-Spruck and Chen-Giga-Goto. We establish two special cases of the comparison principle, existence, uniqueness and basic geometric properties of the flow.