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→ 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 . 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 → ∞.
Capogna, L., Citti, G., & Manfredini, M. (2013). Uniform Gaussian bounds for subelliptic heat kernels and an application to the total variation flow of graphs over Carnot groups. Analysis and Geometry in Metric Spaces, 1, 255-275. https://doi.org/10.2478/agms-2013-0006
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