Sub-Riemannian heat kernels and mean curvature flow of graphs

TL;DR

This paper develops a sub-Riemannian version of a diffusion algorithm to model horizontal mean curvature flow of graphs in Carnot groups, extending classical Euclidean methods to a more general geometric setting.

Contribution

It introduces a novel sub-Riemannian diffusion algorithm and establishes its connection to horizontal mean curvature flow, using nonlinear semi-group theory.

Findings

Established a sub-Riemannian diffusion algorithm for graphs

Proved the algorithm converges to weak solutions of mean curvature flow

Linked heat flow of characteristic functions to horizontal mean curvature

Abstract

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.