Emre Baspinar

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.

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.

Bence-Merriman-Osher algorithm computes numerically mean curvature flow via solutions of heat equation iteratively initialized at the end of each short time interval. Inspired by the convergence proof of Evans of this algorithm where he employed nonlinear semigroup theory and level set approach to mean curvature flow, we prove here that a similar procedure, but now with a different definition of evolved surface at the end of each short time interval, results in a motion by curvature. The interest of this version of the algorithm is its application corresponding to model of the visual cortex.