Asymptotics for optimal controls for horizontal mean curvature flow

TL;DR

This paper studies the asymptotic behavior of optimal controls near characteristic points in the mean curvature flow within the Heisenberg group, highlighting differences from Euclidean cases.

Contribution

It introduces a control-theoretical framework to analyze singularities in degenerate geometries like the Heisenberg group, focusing on asymptotics near characteristic points.

Findings

Asymptotic analysis of controls near characteristic points

Identification of unique singularities in degenerate geometries

Extension of control methods to non-Euclidean mean curvature flow

Abstract

The solutions to surface evolution problems like mean curvature flow can be expressed as value functions of suitable stochastic control problems, obtained as limit of a family of regularised control problems. The control-theoretical approach is particularly suited for such problems for degenerate geometries like the Heisenberg group. In this situation a new type of singularities absent for the Euclidean mean curvature flow occurs, the so-called characteristic points. This paper investigates the asymptotic behaviour of the regularised optimal controls in the vicinity of such characteristic points.