# 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

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topLuca 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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