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Analysis and Geometry in Metric Spaces

Ed. by Ritoré, Manuel

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Uniform Gaussian Bounds for Subelliptic Heat Kernels and an Application to the Total Variation Flow of Graphs over Carnot Groups

Luca Capogna
  • Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609, USA
  • Dipartimento di Matematica, Piazza Porta S. Donato 5, 40126 Bologna, Italy
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/ Giovanna Citti / Maria Manfredini
Published Online: 2013-08-06 | DOI: https://doi.org/10.2478/agms-2013-0006


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 → ∞.

Keywords: Mean curvature flow; sub-Riemannian geometry; Carnot groups

MSC: 53C44; 53C17; 35R03

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About the article

Received: 2012-12-29

Accepted: 2013-06-26

Published Online: 2013-08-06

Citation Information: Analysis and Geometry in Metric Spaces, Volume 1, Pages 255–275, ISSN (Online) 2299-3274, DOI: https://doi.org/10.2478/agms-2013-0006.

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