We study the problem of accessibility of boundary points for domains in the sub–Riemannian setting of the first Heisenberg group. A sufficient condition for accessibility is given. It is a Dini–type continuity condition for the horizontal gradient of the defining function. The sharpness of this condition is shown by examples.

We investigate the minimal surface problem in the three dimensional Heisenberg group, H , equipped with its standard Carnot-Carathéodory metric. Using a particular surface measure, we characterize minimal surfaces in terms of a sub-elliptic partial differential equation and prove an existence result for the Plateau problem in this setting. Further, we provide a link between our minimal surfaces...

We present a notion of frame multiresolution analysis on the Heisenberg group, abbreviated by FMRA, and study its properties. Using the irreducible representations of this group, we shall define a sinc-type function which is our starting point for obtaining the scaling function. Further, we shall give a concrete example of a wavelet FMRA on the Heisenberg group which is analogous to the Shannon...

We construct categorical braid group actions from 2-representations of a Heisenberg algebra. These actions are induced by certain complexes which generalize spherical (Seidel-Thomas) twists and are reminiscent of the Rickard complexes defined by Chuang-Rouquier. Conjecturally, one can relate our complexes to Rickard complexes using categorical vertex operators.

I provide a background of groups viewed as metric spaces to introduce the notion of asymptotic dimension of a group. I analyze the asymptotic dimension of Z ⊕ Z and the free group on two generators to better understand the concept of asymptotic dimension. The asymptotic dimension of the Heisenberg group, H, asdimH, has been shown to be three using advanced mathematics. I will try to show that a...

Jacobians of maps on the Heisenberg group are shown to map suitable group Sobolev spaces into the group Hardy space H1. From this result and a weak∗ convergence theorem for the Hardy space H1 of the Heisenberg group, a compensated compactness property for these Jacobians is obtained. 0. Introduction We investigate compensated compactness properties of Jacobians of maps on the Heisenberg group a...

It is well known that the system ∂xf = a, ∂yf = b on R 2 has a solution if and only if the closure condition ∂xb = ∂ya holds. In this case the solution f is the work done by the force U = (a, b) from the origin to the point (x, y). This paper deals with a similar problem, where the vector fields ∂x, ∂y are replaced by the Heisenberg vector fields X1, X2. In this case the subRiemannian system X1...

If an integrable function f on the Heisenberg group is supported on the set B × R where B ⊂ Cn is compact and the group Fourier transform f̂(λ) is a finite rank operator for all λ ∈ R \ {0}, then f ≡ 0.

The goal of these notes is to introduce the reader to the Heisenberg group with its CarnotCarathéodory metric and to Pansu’s differentiation theorem. As they are very similar, we will first study Rademacher’s theorem about Lipschitz maps and then see how the same technique can be applied in the more complex setting of the Heisenberg group.

We prove the existence of solutions to Monge’s transport problem between two compactly supported Borel probability measures in the Heisenberg group equipped with its Carnot-Carathéodory distance assuming that the initial measure is absolutely continuous with respect to the Haar measure of the group.