Analysis and Geometry in Metric Spaces
In this paper we study heat kernels associated with a Carnot group G, endowed with a family of collapsing left-invariant Riemannian metrics σϵ which converge in the Gromov- Hausdorff sense to a sub-Riemannian structure on G as ϵ rarr; 0. The main new contribution are Gaussian-Type bounds on the heat kernel for the σϵ metrics which are stable as ϵ rarr; 0 and extend the previous time-independent estimates in . As an application we study well posedness of the total variation flow of graph surfaces over a bounded domain in a step two Carnot group (G; σϵ ). We establish interior and boundary gradient estimates, and develop a Schauder theory which are stable as ϵ rarr; 0. As a consequence we obtain long time existence of smooth solutions of the sub-Riemannian flow (ϵ = 0), which in turn yield sub-Riemannian minimal surfaces as rarr;.
Carnot groups, Mean curvature flow, Sub-riemannian geometry
Capogna, Luca; Citti, Giovanna; and Manfredini, Maria, "Uniform Gaussian Bounds for Subelliptic Heat Kernels and an Application to the Total Variation Flow of Graphs Over Carnot Groups" (2013). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.