Document Type

Article

Publication Date

1-1-2013

Publication Title

Analysis and Geometry in Metric Spaces

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 → ∞.

Volume

1

First Page Number

255

Last Page Number

275

DOI

10.2478/agms-2013-0006

Version

Published version

Rights

Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.

Included in

Mathematics Commons

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