  We introduce a sub-Riemannian analogue of the Bence-Merriman-Osher diffusion
driven algorithm and show that it leads to weak solutions of the horizontal
mean curvature flow of graphs over sub-Riemannian Carnot groups. The proof
follows the nonlinear semi-group theory approach originally introduced by L. C.
Evans in the Euclidean setting and is based on new results on the relation
between sub-Riemannian heat flows of characteristic functions of subgraphs and
the horizontal mean curvature of the corresponding graphs.
