Heisenberg Group H1 Research Articles

Criterion for ellipticity on Heisenberg group

We provide a semi-constructive criterion for ellipticity of the differential operator on the Heisenberg group H1.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}^1.$$\\end{document}

CMC and CGC surfaces in Lorentz Heisenberg group H3

The aim of this paper is to study the characterization of some constant mean curvature (CMC) and constant Gauss curvature (CGC) A-net surfaces in the 3-dimensional Lorentz-Heisenberg group H3 endowed with a left invariant Lorentzian metric gi in the three following cases : 1. g1 = -dx2 + dy2 + (xdy + dz)2, 2. g2 = dx2 + dy2 - (xdy + dz)2. 3. g3 = dx2 + (xdy + dz)2 - [(1 - x)dy - dz]2. Using Levi-Civita connection, we will calculate the mean curvature H and the Gaussian curvature K in order to study CMC and CGC A-net surfaces in (H3, g1), (H3, g2) and (H3, g3).

Hardy spaces and quasiconformal maps in the Heisenberg group

We define Hardy spaces Hp, 0<p<∞, for quasiconformal mappings on the Korányi unit ball B in the first Heisenberg group H1. Our definition is stated in terms of the Heisenberg polar coordinates introduced by Korányi and Reimann, and Balogh and Tyson. First, we prove the existence of p0(K)>0 such that every K-quasiconformal map f:B→f(B)⊂H1 belongs to Hp for all 0<p<p0(K). Second, we give two equivalent conditions for the Hp membership of a quasiconformal map f, one in terms of the radial limits of f, and one using a nontangential maximal function of f. As an application, we characterize Carleson measures on B via integral inequalities for quasiconformal mappings on B and their radial limits. Our paper thus extends results by Astala and Koskela, Jerison and Weitsman, Nolder, and Zinsmeister, from Rn to H1. A crucial difference between the proofs in Rn and H1 is caused by the nonisotropic nature of the Korányi unit sphere with its two characteristic points.

A characterization of constant p-mean curvature surfaces in the Heisenberg group H1

In Euclidean 3-space, it is well known that the Sine-Gordon equation was considered in the nineteenth century in the course of investigations of surfaces of constant Gaussian curvature K=−1. Such a surface can be constructed from a solution to the Sine-Gordon equation, and vice versa. With this as motivation, employing the fundamental theorem of surfaces in the Heisenberg group H1, we show in this paper that the existence of a constant p-mean curvature surface (without singular points) is equivalent to the existence of a solution to a nonlinear second-order ODE (1.2), which is a kind of Liénard equations. Therefore, we turn to investigate this equation. It is a surprise that we give a complete set of solutions to (1.2) (or (1.5)), and hence use the types of the solution to divide constant p-mean curvature surfaces into several classes. As a result, after a kind of normalization, we obtain a representation of constant p-mean curvature surfaces and classify further all constant p-mean curvature surfaces. In Section 9, we provide an approach to construct p-minimal surfaces. It turns out that, in some sense, generic p-minimal surfaces can be constructed via this approach. Finally, as a derivation, we recover the Bernstein-type theorem which was first shown in [14] (or see [19,20]).

On rectifiable measures in Carnot groups: Marstrand–Mattila rectifiability criterion

In this paper we continue the study of the notion of P-rectifiability in Carnot groups. We say that a Radon measure is Ph-rectifiable, for h∈N, if it has positive h-lower density and finite h-upper density almost everywhere, and, at almost every point, it admits a unique tangent measure up to multiples.In this paper we prove a Marstrand–Mattila rectifiability criterion in arbitrary Carnot groups for P-rectifiable measures with tangent planes that admit a normal complementary subgroup. Namely, in this co-normal case, even if a priori the tangent planes at a point might not be the same at different scales, a posteriori the measure has a unique tangent almost everywhere.Since every horizontal subgroup of a Carnot group has a normal complement, our criterion applies in the particular case in which the tangents are one-dimensional horizontal subgroups. Hence, as an immediate consequence of our Marstrand–Mattila rectifiability criterion and a result of Chousionis–Magnani–Tyson, we obtain the one-dimensional Preiss's theorem in the first Heisenberg group H1. More precisely, we show that a Radon measure ϕ on H1 with positive and finite one-density with respect to the Koranyi distance is absolutely continuous with respect to the one-dimensional Hausdorff measure H1, and it is supported on a one-rectifiable set in the sense of Federer, i.e., it is supported on the countable union of the images of Lipschitz maps from A⊆R to H1.

Classification of space-like translation surfaces in the 3-dimensional Lorentz Heisenberg group H3

PurposeIn the Lorentz Heisenberg space H3 endowed with flat metric g3, a translation surface is parametrized by r(x, y) = γ1(x)*γ2(y), where γ1 and γ2 are two planar curves lying in planes, which are not orthogonal. In this article, we classify translation surfaces in H3, which satisfy some algebraic equations in terms of the coordinate functions and the Laplacian operator with respect to the first fundamental form of the surface.Design/methodology/approachIn this paper, we classify some type of space-like translation surfaces of H3 endowed with flat metric g3 under the conditionΔri = λiri. We will develop the system which describes surfaces of type finite in H3. For solve the system thus obtained, we will use the calculation variational. Finally, we will try to give performances geometric surfaces that meet the condition imposed.FindingsClassification of six types of translation surfaces of finite type in the three-dimensional Lorentz Heisenberg group H3.Originality/valueThe subject of this paper lies at the border of geometry differential and spectral analysis on manifolds. Historically, the first research on the study of sub-finite type varieties began around the 1970 by B.Y.Chen. The idea was to find a better estimate of the mean total curvature of a compact subvariety of a Euclidean space. In fact, the notion of finite type subvariety is a natural extension of the notion of a minimal subvariety or surface, a notion directly linked to the calculation of variations. The goal of this work is the classification of surfaces in H3, in other words the surfaces which satisfy the condition/Delta (ri) = /Lambda (ri), such that the Laplacian is associated with the first, fundamental form.

Pansu–Wulff shapes in ℍ1

Abstract We consider an asymmetric left-invariant norm ∥ ⋅ ∥ K {\|\cdot\|_{K}} in the first Heisenberg group ℍ 1 {\mathbb{H}^{1}} induced by a convex body K ⊂ ℝ 2 {K\subset\mathbb{R}^{2}} containing the origin in its interior. Associated to ∥ ⋅ ∥ K {\|\cdot\|_{K}} there is a perimeter functional, that coincides with the classical sub-Riemannian perimeter in case K is the closed unit disk centered at the origin of ℝ 2 {{\mathbb{R}}^{2}} . Under the assumption that K has C 2 {C^{2}} boundary with strictly positive geodesic curvature we compute the first variation formula of perimeter for sets with C 2 {C^{2}} boundary. The localization of the variational formula in the non-singular part of the boundary, composed of the points where the tangent plane is not horizontal, allows us to define a mean curvature function H K {H_{K}} out of the singular set. In the case of non-vanishing mean curvature, the condition that H K {H_{K}} be constant implies that the non-singular portion of the boundary is foliated by horizontal liftings of translations of ∂ ⁡ K {\partial K} dilated by a factor of 1 H K {\frac{1}{H_{K}}} . Based on this we can define a sphere 𝕊 K {\mathbb{S}_{K}} with constant mean curvature 1 by considering the union of all horizontal liftings of ∂ ⁡ K {\partial K} starting from ( 0 , 0 , 0 ) {(0,0,0)} until they meet again in a point of the vertical axis. We give some geometric properties of this sphere and, moreover, we prove that, up to non-homogeneous dilations and left-translations, they are the only solutions of the sub-Finsler isoperimetric problem in a restricted class of sets.

Multiparameter singular integrals on the Heisenberg group: uniform estimates

We consider a class of multiparameter singular Radon integral operators on the Heisenberg group H 1 {\mathbb H}^1 where the underlying submanifold is the graph of a polynomial. A remarkable difference with the euclidean case, where Heisenberg convolution is replaced by euclidean convolution, is that the operators on the Heisenberg group are always L 2 L^2 bounded. This is not the case in the euclidean setting where L 2 L^2 boundedness depends on the polynomial defining the underlying surface. Here we uncover some new, interesting phenomena. For example, although the Heisenberg group operators are always L 2 L^2 bounded, the bounds are not uniform in the coefficients of polynomials with fixed degree. When we ask for which polynomials uniform L 2 L^2 bounds hold, we arrive at the same class where uniform bounds hold in the euclidean case.

The Wiener Measure on the Heisenberg Group and Parabolic Equations

In this paper, we study questions related to the theory of stochastic processes on Lie nilpotent groups. In particular, we consider the stochastic process on the Heisenberg group H3(ℝ) whose trajectories satisfy the horizontal conditions in the stochastic sense relative to the standard contact structure on H3 (ℝ). It is shown that this process is a homogeneous Markov process relative to the Heisenberg group operation. There was found a representation in the form of a Wiener integral for a one-parameter linear semigroup of operators for which the Heisenberg sublaplacian generated by basis vector fields of the corresponding Lie algebra L(H3) is producing. The main method of solving the problem in this paper is using the path integrals technique, which indicates the common direction of further development of the results.

Solution of the Cauchy Problem for the Heat Equation on the Heisenberg Group and the Wiener Integral

The issues related to applications of functional integrals to evolution equations are studied. In particular, this is the problem of representation of solutions to the Cauchy problem for the heat equation in the three-parameter Heisenberg group H3(ℝ) in terms of Wiener integral in the space of trajectories from C[0, t] × C[0, t].

On the Transfer of the Wiener Measure to the Set of Continuous Trajectories in the Heisenberg Group

In the paper, problems related to the theory of stochastic processes on nilpotent Lie groups are studied. In particular, a stochastic process on the Heisenberg group H3(ℝ) is considered such that the trajectories of this process, in the stochastic sense, satisfy the horizontality conditions with respect to the standard contact structure on H3(ℝ). The main result claims that the measure defined on the trajectories of this process is completely concentrated on the set C([0, t], H3(ℝ)) of continuous trajectories.

Some New Characterizations of Parallel Factorable Surface in Riemannian Three Dimensional Heisenberg Group

Bu çalışmada üç boyutlu Riemannian Heisenberg grubunda grup işlemiyle elde edilen paralel faktorable yüzeylerin bazı özellikleri verildi. Daha sonra H₃ te Levi- Civita konneksiyonlarına göre paralel faktorable yüzeylerin bazı karakterizasyonları elde edildi.

CMC Spheres in the Heisenberg Group

Abstract We study a family of spheres with constant mean curvature (CMC) in the Riemannian Heisenberg group H 1. These spheres are conjectured to be the isoperimetric sets of H 1. We prove several results supporting this conjecture. We also focus our attention on the sub-Riemannian limit.

Stability of group relations under small Hilbert–Schmidt perturbations

If matrices almost satisfying a group relation are close to matrices exactly satisfying the relation, then we say that a group is matricially stable. Here “almost” and “close” are in terms of the Hilbert–Schmidt norm. Using tracial 2-norm on II1-factors we similarly define II1-factor stability for groups. Our main result is that all 1-relator groups with non-trivial center are II1-factor stable. Many of them are also matricially stable and RFD. For amenable groups we give a complete characterization of matricial stability in terms of the following approximation property for characters: each character must be a pointwise limit of traces of finite-dimensional representations. This allows us to prove matricial stability for the discrete Heisenberg group H3 and for all virtually abelian groups. For non-amenable groups the same approximation property is a necessary condition for being matricially stable. We study this approximation property and show that RF groups with character rigidity have it.

On the 𝐶^{1,𝛼} regularity of 𝑝-harmonic functions in the Heisenberg group

We present a proof of the local Hölder regularity of the horizontal derivatives of weak solutions to the p p -Laplace equation in the Heisenberg group H 1 \mathbb {H}^1 for p &gt; 4 p&gt;4 .

Area-stationary and stable surfaces of class [formula omitted] in the sub-Riemannian Heisenberg group [formula omitted

We consider surfaces of class C1 in the 3-dimensional sub-Riemannian Heisenberg group H1. Assuming the surface is area-stationary, i.e., a critical point of the sub-Riemannian perimeter under compactly supported variations, we show that its regular part is foliated by horizontal straight lines. In case the surface is complete and oriented, without singular points, and stable, i.e., a second order minimum of perimeter, we prove that the surface must be a vertical plane. This implies the following Bernstein type result: a complete locally area-minimizing intrinsic graph of a C1 function in H1 is a vertical plane.

Classification of left-invariant metrics on the Heisenberg group

In this study, we investigate the Riemannian and Lorentzian geometry of left-invariant metrics on the Heisenberg group H2n+1, of dimension 2n+1. We describe the space of all the left-invariant metrics of Riemannian and Lorentzian signatures up to automorphisms of the Heisenberg group. Thus, we classify quadratic forms of the corresponding signatures with respect to the action of the symplectic group. We also investigate the curvature properties and holonomy of these metrics. The most interesting is the Lorentzian metric with a parallel, null, central, left-invariant vector field. Rahmani proved that this metric is flat in the case of Heisenberg group H3. We show that this metric is not flat in higher dimensions.

Some Weighted Group Algebras are Operator Algebras

AbstractLet G be a finitely generated group with polynomial growth, and let ω be a weight, i.e. a sub-multiplicative function on G with positive values. We study when the weighted group algebra ℓ1 (G, ω) is isomorphic to an operator algebra. We show that ℓ1 (G, ω) is isomorphic to an operator algebra if ω is a polynomial weight with large enough degree or an exponential weight of order 0 &lt; α &lt; 1. We demonstrate that the order of growth of G plays an important role in this problem. Moreover, the algebraic centre of ℓ1 (G, ω) is isomorphic to a Q-algebra, and hence satisfies a multi-variable von Neumann inequality. We also present a more detailed study of our results when G consists of the d-dimensional integers ℤd or the three-dimensional discrete Heisenberg group ℍ3(ℤ). The case of the free group with two generators is considered as a counter-example of groups with exponential growth.

Pseudo-Riemannian Symmetries on Heisenberg Groups

Abstract The notion of Γ-symmetric space is a natural generalization of the classical notion of symmetric space based on Z2-grading on Lie algebras. We consider homogeneous spaces G/H such that the Lie algebra g of G admits a Γ-grading where Γ is a finite abelian group. In this work we study Riemannian metrics and Lorentzian metrics on the Heisenberg group H3 adapted to the symmetries of a Γ-symmetric structure on H3. We prove that the classification of Riemannian and Lorentzian Zl-symmetric metrics on H3 corresponds to the classification of its left-invariant Riemannian and Lorentzian metrics, up to isometry. We study also the Z§-symmetric structures on G/H when G is the (2p + 1)-dimensional Heisenberg group. This gives examples of non-Riemannian symmetric spaces. When k &gt; 1, we show that there exists a family of flat and torsion free affine connections adapted to the Z§-symmetric structures.

Algebraically determined semidirect products

Let G be a Polish (i.e., complete separable metric topological) group. Define G to be an algebraically determined Polish group if given any Polish group L and an algebraic isomorphism φ:L↦G, then φ is a topological isomorphism. The purpose of this paper is to prove a general theorem that gives useful sufficient conditions for a semidirect product of two Polish groups to be algebraically determined. This general theorem will provide a flowchart or recipe for proving that some special semidirect products are algebraically determined. For example, it may be used to prove that the natural semidirect product H⋊G, where H is the additive group of a separable Hilbert space and G is a Polish group of unitaries on H acting transitively on the unit sphere with −I∈G, is algebraically determined. An example of such a G is the unitary group of a separable irreducible C⁎-algebra with identity on H. Not all nontrivial semidirect products of Polish groups are algebraically determined, for it is known that the Heisenberg group H3(R) is a semidirect product of the form R2⋊θR1 and is not an algebraically determined Polish group.