Geodesic
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, Tome 1 (2013) no. 1, pp. 255-275.

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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 → ∞.
Mots-clés : Mean curvature flow, sub-Riemannian geometry, Carnot groups 
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     title = {Uniform {Gaussian} {Bounds} for {Subelliptic} {Heat} {Kernels} and an {Application} to the {Total} {Variation} {Flow} of {Graphs} over {Carnot} {Groups}},
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Capogna, Luca; Citti, Giovanna; Manfredini, Maria. 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, Tome 1 (2013) no. 1, pp. 255-275. http://geodesic.mathdoc.fr/item/AGMS_2013_1_1_a11/