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

• Autorzy: Luca Capogna , Giovanna Citti , Maria Manfredini
• Języki publikacji: EN

Abstrakty

EN

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 → ∞.

Słowa kluczowe

EN

Mean curvature flow; sub-Riemannian geometry; Carnot groups

Kategorie tematyczne

• 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
• 35R03: Partial differential equations on Heisenberg groups, Lie groups, Carnot groups, etc.
• 53C17: Sub-Riemannian geometry

• Wydawca: De Gruyter Open
• Czasopismo: Analysis and Geometry in Metric Spaces
• Rocznik: 2013
• Tom: 1
• Strony: 255-275

Daty

otrzymano

2012-12-29

zaakceptowano

2013-06-26

online

2013-08-06

Twórcy

autor

Luca Capogna

• Department of Mathematical Sciences,
  Worcester Polytechnic Institute, Worcester, MA 01609, USA
• Dipartimento di Matematica, Piazza Porta S. Donato 5,
  40126 Bologna, Italy

autor

Giovanna Citti

• Dipartimento di Matematica, Piazza Porta S. Donato 5,
  40126 Bologna, Italy

autor

Maria Manfredini

• Dipartimento di Matematica, Piazza Porta S. Donato 5,
  40126 Bologna, Italy

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Identyfikatory

DOI

10.2478/agms-2013-0006