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 → ∞.

Creator

• Citti, Giovanna
• Capogna, Luca
• Manfredini, Maria

Keyword

• Mean Curvature Flow
• Carnot Groups
• Sub-Riemannian Geometry

Date created

• 1/1/13

Resource type

• Article

Extent

• 21 pages

Source

• https://doi.org/10.2478/agms-2013-0006

Rights statement

• In Copyright

Last modified

• 2020-09-22