Gradient estimates for the heat kernels in higher dimensional Heisenberg groups

2010 ◽  
Vol 31 (3) ◽  
pp. 305-314 ◽  
Author(s):  
Bin Qian
Keyword(s):  
Heat Kernels ◽  
Gradient Estimates ◽  
Heisenberg Groups ◽  
Higher Dimensional

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

2013 ◽  
Vol 1 ◽  
pp. 255-275 ◽  
Author(s):  
Luca Capogna ◽  
Giovanna Citti ◽  
Maria Manfredini
Keyword(s):  
Total Variation ◽  
Carnot Group ◽  
Riemannian Metrics ◽  
Heat Kernels ◽  
Gradient Estimates ◽  
Long Time Existence ◽  
Total Variation Flow ◽  
Long Time ◽  
Invariant Riemannian Metrics ◽  
Time Existence

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 → ∞.

Heat kernels and Green functions of sub-Laplacians on Heisenberg groups with multi-dimensional center

2018 ◽  
Vol 13 (4) ◽  
pp. 38
Author(s):  
Shahla Molahajloo ◽  
M.W. Wong
Keyword(s):  
Fourier Transform ◽  
Green Function ◽  
Heisenberg Group ◽  
Inverse Fourier Transform ◽  
Heat Kernels ◽  
Hermite Functions ◽  
Green Functions ◽  
Heisenberg Groups ◽  
Explicit Formulas ◽  
Special Hermite Functions

We compute the sub-Laplacian on the Heisenberg group with multi-dimensional center. By taking the inverse Fourier transform with respect to the center, we get the parametrized twisted Laplacians. Then by means of the special Hermite functions, we find the eigenfunctions and the eigenvalues of the twisted Laplacians. The explicit formulas for the heat kernels and Green functions of the twisted Laplacians can then be obtained. Then we give an explicit formula for the heat kernal and Green function of the sub-Laplacian on the Heisenberg group with multi-dimensional center.

scholarly journals Gradient estimates of Dirichlet heat kernels for unimodal Lévy processes

2017 ◽  
Vol 291 (2-3) ◽  
pp. 374-397 ◽  
Author(s):  
Tadeusz Kulczycki ◽  
Michał Ryznar
Keyword(s):  
Lévy Processes ◽  
Levy Processes ◽  
Heat Kernels ◽  
Gradient Estimates

scholarly journals Complex tangencies to embeddings of Heisenberg groups and odd-dimensional spheres

2014 ◽  
Vol 25 (03) ◽  
pp. 1450028
Author(s):  
Ali M. Elgindi
Keyword(s):  
Heisenberg Group ◽  
Topological Structure ◽  
Dimensional Sphere ◽  
Heisenberg Groups ◽  
3 Dimensional ◽  
Complex Spaces ◽  
Higher Dimensional ◽  
Complex Tangent

The notion of a complex tangent arises for embeddings of real manifolds into complex spaces. It is of particular interest when studying embeddings of real n-dimensional manifolds into ℂn. The generic topological structure of the set complex tangents to such embeddings Mn ↪ ℂn takes the form of a (stratified) (n-2)-dimensional submanifold of Mn. In this paper, we generalize our results from our previous work for the 3-dimensional sphere and the Heisenberg group to obtain results regarding the possible topological configurations of the sets of complex tangents to embeddings of odd-dimensional spheres S2n-1 ↪ ℂ2n-1 by first considering the situation for the higher-dimensional analogues of the Heisenberg group.

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