Regularity of mean curvature flow of graphs on Lie groups free up to step 2

TL;DR

This paper proves uniform regularity and long-term existence of mean curvature flow of graphs on step 2 Lie groups, including subRiemannian limits, extending previous methods to broader geometric settings.

Contribution

It establishes uniform $C^{k,eta}$ estimates and long-time existence for mean curvature flow on step 2 Lie groups, including non-nilpotent cases, and extends techniques to total variation flow.

Findings

Uniform $C^{k,eta}$ estimates as $ ext{e} o 0$

Long-time existence of the flow in subRiemannian limit

Extension to all step two Carnot groups

Abstract

We consider (smooth) solutions of the mean curvature flow of graphs over bounded domains in a Lie group free up to step two (and not necessarily nilpotent), endowed with a one parameter family of Riemannian metrics $\sigma_\e$ collapsing to a subRiemannian metric $\sigma_0$ as $\e\to 0$. We establish $C^{k,\alpha}$ estimates for this flow, that are uniform as $\e\to 0$ and as a consequence prove long time existence for the subRiemannian mean curvature flow of the graph. Our proof extend to the setting of every step two Carnot group (not necessarily free) and can be adapted following our previous work in \cite{CCM3} to the total variation flow.