Sub-Riemannian heat kernels and mean curvature flow of graphs

We introduce a sub-Riemannian analogue of the Bence–Merriman–Osher algorithm (Merriman et al., 1992 [42], ) 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 (1993) [27] 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.