Uniqueness of viscosity mean curvature flow solution in two sub-Riemannian structures

TL;DR

This paper establishes the uniqueness of viscosity solutions for sub-Riemannian mean curvature flow in specific geometric structures by linking them to Riemannian flows, overcoming limitations of comparison principles.

Contribution

It proves that viscosity solutions can be approximated by Riemannian solutions, ensuring uniqueness in sub-Riemannian settings where comparison principles fail.

Findings

Viscosity solutions are limits of Riemannian solutions.

Uniqueness and comparison principles hold in the approximating Riemannian setting.

Results apply to SE(2) and Carnot groups of step 2.

Abstract

Here we provide a uniqueness result for viscosity solutions to sub-Riemannian mean curvature flow. In this setting the uniqueness cannot be deduced via comparison principle, which is known only for graphs and for radially symmetric surfaces, due to the presence of characteristic points. Here we prove that any viscosity solution is limit of a sequence of solutions of Riemannian flow, and obtain as a consequence uniqueness and comparison principle already known in the approximating riemannian setting. The results are provided in the settings of both 3-dimensional rototranslation group SE(2) and Carnot groups of step 2, which are particularly important due to their relation to surface completion problem of model of the visual cortex.