Uniform Gaussian bounds for subelliptic heat kernels and an application to the total variation flow of graphs over Carnot groups

TL;DR

This paper establishes stable Gaussian bounds for heat kernels on Carnot groups with collapsing Riemannian metrics and applies these results to analyze the total variation flow of graphs, leading to existence of minimal surfaces.

Contribution

It provides new Gaussian-type bounds on heat kernels that are stable under metric collapse and applies these bounds to the well-posedness of the total variation flow in sub-Riemannian settings.

Findings

Stable Gaussian bounds for heat kernels as metrics collapse

Well-posedness and gradient estimates for total variation flow

Existence of sub-Riemannian minimal surfaces as time progresses

Abstract

In this paper we study heat kernels associated to a Carnot group $G$, endowed with a family of collapsing left-invariant Riemannian metrics $\sigma_\e$ which converge in the Gromov-Hausdorff sense to a sub-Riemannian structure on $G$ as $\e\to 0$. The main new contribution are Gaussian-type bounds on the heat kernel for the $\sigma_\e$ metrics which are stable as $\e\to 0$ and extend the previous time-independent estimates in \cite{CiMa-F}. As an application we study well posedness of the total variation flow of graph surfaces over a bounded domain in $(G,\s_\e)$. We establish interior and boundary gradient estimates, and develop a Schauder theory which are stable as $\e\to 0$. As a consequence we obtain long time existence of smooth solutions of the sub-Riemannian flow ($\e=0$), which in turn yield sub-Riemannian minimal surfaces as $t\to \infty$.