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

In this paper we study heat kernels associated to a Carnot group $G$, endowed with a family of collapsing left-invariant Riemannian metrics ${\sigma }_{\text{e}}$ which converge in the Gromov-Hausdorff sense to a sub-Riemannian structure on $G$ as $\text{e}\to 0$. The main new contribution are Gaussian-type bounds on the heat kernel for the ${\sigma }_{\text{e}}$ metrics which are stable as $\text{e}\to 0$ 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 $(G,{\text{s}}_{\text{e}})$. We establish interior and boundary gradient estimates, and develop a Schauder theory which are stable as $\text{e}\to 0$. As a consequence we obtain long time existence of smooth solutions of the sub-Riemannian flow ($\text{e}=0$), which in turn yield sub-Riemannian minimal surfaces as $t\to \mathrm{\infty }$.