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

Luca Capogna; Giovanna Citti; Maria Manfredini

Analysis and Geometry in Metric Spaces (2013)

  • Volume: 1, page 255-275
  • ISSN: 2299-3274

Abstract

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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 → ∞.

How to cite

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Luca Capogna, Giovanna Citti, and Maria Manfredini. "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 (2013): 255-275. <http://eudml.org/doc/266636>.

@article{LucaCapogna2013,
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 → ∞.},
author = {Luca Capogna, Giovanna Citti, Maria Manfredini},
journal = {Analysis and Geometry in Metric Spaces},
keywords = {Mean curvature flow; sub-Riemannian geometry; Carnot groups; mean curvature flow; sub-Riemannian minimal surfaces},
language = {eng},
pages = {255-275},
title = {Uniform Gaussian Bounds for Subelliptic Heat Kernels and an Application to the Total Variation Flow of Graphs over Carnot Groups},
url = {http://eudml.org/doc/266636},
volume = {1},
year = {2013},
}

TY - JOUR
AU - Luca Capogna
AU - Giovanna Citti
AU - Maria Manfredini
TI - Uniform Gaussian Bounds for Subelliptic Heat Kernels and an Application to the Total Variation Flow of Graphs over Carnot Groups
JO - Analysis and Geometry in Metric Spaces
PY - 2013
VL - 1
SP - 255
EP - 275
AB - 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 → ∞.
LA - eng
KW - Mean curvature flow; sub-Riemannian geometry; Carnot groups; mean curvature flow; sub-Riemannian minimal surfaces
UR - http://eudml.org/doc/266636
ER -

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