On the horizontal Mean Curvature Flow for Axisymmetric surfaces in the Heisenberg Group

TL;DR

This paper investigates the horizontal mean curvature flow of axisymmetric surfaces in the Heisenberg group using the level-set method, establishing key properties and providing explicit solutions.

Contribution

It introduces a framework for analyzing the flow in the Heisenberg group, proving existence, uniqueness, and stability of solutions, and offering explicit solutions for specific initial surfaces.

Findings

Proved existence and uniqueness of solutions

Established stability of the flow

Provided explicit solution for a subelliptic sphere

Abstract

We study the horizontal mean curvature flow in the Heisenberg group by using the level-set method. We prove the uniqueness, existence and stability of axisymmetric viscosity solutions of the level-set equation. An explicit solution is given for the motion starting from a subelliptic sphere. We also give several properties of the level-set method and the mean curvature flow in the Heisenberg group.