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

TL;DR

This paper proves the uniqueness of vanishing viscosity solutions for sub-Riemannian mean curvature flow in specific structures, extending previous results to characteristic points and providing convergence estimates.

Contribution

It introduces a vanishing viscosity approach to establish uniqueness at characteristic points in sub-Riemannian mean curvature flow.

Findings

Uniqueness of solutions in $ ext{SE}(2)$ and $ ext{H}^1$ groups

Convergence rates for approximating solutions

Relevance to visual cortex surface completion

Abstract

Here we provide uniqueness of vanishing viscosity solutions to sub-Riemannian mean curvature flow problem, which was known only far from characteristic points or under special symmetry condition. We employ vanishing viscosity approach and look for solutions as limit of solutions to approximating flow, which is well defined also at characteristic points, and estimate the rate of convergence of the approximating solutions. The results are provided in the settings of both 3-dimensional rototranslation group $\text{SE}(2)$ and Heisenberg group $\text{H}^1$ and they are particularly important due to their relation to surface completion problem of model of the visual cortex.