Abstract
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→
0. The main new contribution are Gaussian-type bounds on
the heat kernel for the σε metrics which are stable as ε→0
and extend the previous time-independent estimates in [16].
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 → 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 t → ∞.
Convergent numerical approximation of the stochastic total variation flow
