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

Abstract

In this paper we study heat kernels associated to a Carnot group $G$, endowed with a family of collapsing left-invariant Riemannian metrics $σ_\e$ 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 $σ_\e$ 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 $t\to \infty$.

We have corrected a few typos and added a few more details to the proof of the Gaussian estimates

References in corpus (5)

• Minimal surfaces in the Heisenberg group
• Minimal surfaces in pseudohermitian geometry
• Area-stationary surfaces in the Heisenberg group H^1
• Regularity of non-characteristic minimal graphs in the Heisenberg group $H^1$
• Smoothness of Lipschitz minimal intrinsic graphs in Heisenberg groups $H^n,n>1$