Dimensional Heisenberg Group Research Articles

Geometry of cotangent bundle of Heisenberg group

In this paper, the classification of left invariant Riemannian metrics on the cotangent bundle of the (2n+1)-dimensional Heisenberg group up to the action of the automorphism group is presented. Moreover, it is proved that the complex structure on this group is unique, and the corresponding pseudo-Kähler metrics are described and shown to be Ricci flat. It is known that this algebra admits an ad-invariant metric of neutral signature. Here, the uniqueness of such metric is proved.

A classification of left-invariant Lorentzian metrics on some nilpotent Lie groups

It has been known that there exist exactly three left-invariant Lorentzian metrics up to scaling and automorphisms on the three dimensional Heisenberg group. In this paper, we classify left-invariant Lorentzian metrics on the direct product of three dimensional Heisenberg group and the Euclidean space of dimension $n-3$ with $n \geq 4$, and prove that there exist exactly six such metrics on this Lie group up to scaling and automorphisms. Moreover we show that only one of them is flat, and the other five metrics are Ricci solitons but not Einstein. We also characterize this flat metric as the unique closed orbit, where the equivalence class of each left-invariant metric can be identified with an orbit of a certain group action on some symmetric space.

Branching rule and coadjoint orbit for Heisenberg Gelfand pairs

Let [Formula: see text] be a closed connected subgroup of the unitary group [Formula: see text] and let [Formula: see text] be the [Formula: see text]-dimensional Heisenberg group ([Formula: see text]). We consider the semidirect product [Formula: see text] such that [Formula: see text] is a Gelfand pair. Let [Formula: see text] be the respective Lie algebras of [Formula: see text] and [Formula: see text] and [Formula: see text] be the natural projection. It was pointed out by Lipsman, that the unitary dual [Formula: see text] of [Formula: see text] is in one-to-one correspondence with the space of admissible coadjoint orbits [Formula: see text] (see [10]). Let [Formula: see text] be a generic representation of [Formula: see text] and let [Formula: see text]. To these representations we associate, respectively, the admissible coadjoint orbit [Formula: see text] and [Formula: see text] (via the Lipsman’s correspondence). We denote by [Formula: see text] the number of [Formula: see text]-orbits in [Formula: see text] which is called the Corwin–Greenleaf multiplicity function. The Kirillov–Lipsman’s orbit method suggests that the multiplicity [Formula: see text] of an irreducible [Formula: see text]-module [Formula: see text] occurring in the restriction [Formula: see text] could be read from the coadjoint action of [Formula: see text] on [Formula: see text] In this paper, we show that [Formula: see text] For the special case [Formula: see text] (maximal torus in the unitary group [Formula: see text]) and [Formula: see text], we prove that [Formula: see text]

Riemannian and sub-Riemannian structures on a cotangent bundle of Heisenberg group

In this paper we give a classification of left invariant sub-Riemannian structures on cotangent bundle of 2n+1 dimensional Heisenberg group T*H2n+1. We show that the sub-Riemannian metric is tamed by the corresponding Riemannian metric on T*H2n+1. We also describe Riemannian and sub-Riemannian geodesics on T*H2n+1.

Least energy solutions to quasilinear subelliptic equations with constant and degenerate potentials on the Heisenberg group

Let H n = C n × R $\mathbb {H}^{n}=\mathbb {C}^{n}\times \mathbb {R}$ be the n $n$ -dimensional Heisenberg group, Q = 2 n + 2 $Q=2n+2$ be the homogeneous dimension of H n $\mathbb {H}^{n}$ . In this paper, we investigate the existence of a least energy solution to the Q $Q$ -subLaplacian Schrödinger equation with either a constant V = γ $V=\gamma$ or a degenerate potential V $V$ vanishing on a bounded open subset of H n $\mathbb {H}^n$ : − div H ∇ H u Q − 2 ∇ H u + V ( ξ ) u Q − 2 u = f u $$\begin{equation} -\mathrm{div}_{\mathbb {H}}{\left({\left|\nabla _{\mathbb {H}}u\right|}^{Q-2} \nabla _{\mathbb {H}}u\right)} +V(\xi ) {\left|u\right|}^{Q-2}u=f{\left(u\right)} \end{equation}$$ (0.1)with the non-linear term f $f$ of maximal exponential growth exp ( α t Q Q − 1 ) $\exp (\alpha t^{\frac{Q}{Q-1}})$ as t → + ∞ $t\rightarrow +\infty$ . Since the Pólya–Szegö-type inequality fails on H n $\mathbb {H}^n$ , the coercivity of the potential has been a standard assumption in the literature for subelliptic equations to exclude the vanishing phenomena of Palais–Smale sequence on the entire space H n $\mathbb {H}^n$ . Our aim in this paper is to remove this strong assumption. To this end, we first establish a sharp critical Trudinger–Moser inequality involving a degenerate potential on H n $\mathbb {H}^n$ . Second, we prove the existence of a least energy solution to the above equation with the constant potential V ( ξ ) = γ > 0 $V(\xi )=\gamma >0$ . Third, we establish the existence of a least energy solution to the Q $Q$ -subelliptic equation (0.1) involving the degenerate potential which vanishes on some open bounded set of H n $\mathbb {H}^{n}$ . We develop arguments that avoid using any symmetrization on H n $\mathbb {H}^n$ where the Pólya–Szegö inequality fails. Fourth, we also establish the existence of a least energy solution to (0.1) when the potential is a non-degenerate Rabinowitz type potential but still fails to be coercive. Our results in this paper improve significantly on the earlier ones on quasilinear Schrödinger equations on the Heisenberg group in the literature. We note that all the main results and their proofs in this paper hold on stratified groups with the same proofs.

A classification of left-invariant pseudo-Riemannian metrics on some nilpotent Lie groups*

It is known that a connected and simply-connected Lie group admits only one left-invariant Riemannian metric up to scaling and isometry if and only if it is isomorphic to the Euclidean space, the Lie group of the real hyperbolic space, or the direct product of the three dimensional Heisenberg group and the Euclidean space of dimension n−3. In this paper, we give a classification of left-invariant pseudo-Riemannian metrics of an arbitrary signature for the third Lie groups with n≥4 up to scaling and automorphisms. This completes the classifications of left-invariant pseudo-Riemannian metrics for the above three Lie groups up to scaling and automorphisms.

COHOMOGENEITY ONE KÄHLER-RICCI SOLITONS UNDER A HEISENBERG GROUP ACTION AND RELATED METRICS

We find complete expanding Kähler-Ricci solitons of two types. The first are of cohomogeneity one under the action of the (2m - 1)-dimensional Heisenberg group or a certain quotient of it, for any m ≥ 2. The second type, which generalizes the first, reside on ℂ*-bundles over compact Kähler Ricci-flat manifolds, admit an S1 action by isometries, and have two ends. Both types have the local structure stemming from an ansatz first described in [20]. We give curvature bounds for the first type, and asymptotics for both types.

On the Michor–Mumford phenomenon in the infinite dimensional Heisenberg group

In the infinite dimensional Heisenberg group, we construct a left invariant weak Riemannian metric that gives a degenerate geodesic distance. The same construction yields a degenerate sub-Riemannian distance. We show how the standard notion of sectional curvature adapts to our framework, but it cannot be defined everywhere and it is unbounded on suitable sequences of planes. The vanishing of the distance precisely occurs along this sequence of planes, so that the degenerate Riemannian distance appears in connection with an unbounded sectional curvature. In the 2005 paper by Michor and Mumford, this phenomenon was first observed in some specific Fréchet manifolds.

Foliated corona decompositions

We prove that the $L_4$ norm of the vertical perimeter of any measurable subset of the $3$-dimensional Heisenberg group $\mathbb{H}$ is at most a universal constant multiple of the (Heisenberg) perimeter of the subset. We show that this isoperimetric-type inequality is optimal in the sense that there are sets for which it fails to hold with the $L_4$ norm replaced by the $L_q$ norm for any $q<4$. This is in contrast to the $5$-dimensional setting, where the above result holds with the $L_4$ norm replaced by the $L_2$ norm. The proof of the aforementioned isoperimetric inequality introduces a new structural methodology for understanding the geometry of surfaces in $\mathbb{H}$. In previous work (2017) we showed how to obtain a hierarchical decomposition of Ahlfors-regular surfaces into pieces that are approximately intrinsic Lipschitz graphs. Here we prove that any such graph admits a foliated corona decomposition, which is a family of nested partitions into pieces that are close to ruled surfaces. Apart from the intrinsic geometric and analytic significance of these results, which settle questions posed by Cheeger-Kleiner-Naor (2009) and Lafforgue-Naor (2012), they have several noteworthy implications, including the fact that the $L_1$ distortion of a word-ball of radius $n\ge 2$ in the discrete $3$-dimensional Heisenberg group is bounded above and below by universal constant multiples of $\sqrt[4]{\log n}$; this is in contrast to higher dimensional Heisenberg groups, where our previous work showed that the distortion of a word-ball of radius $n\ge 2$ is of order $\sqrt{\log n}$.

Darboux curves in three dimensional Heisenberg group

In this paper, a new characterization for darboux curves in Heis₃ is completely given. Then, a new classification for translation surface , which is generated by darboux curve in Heis₃ is obtained.

On the Cortex of the Groups 𝕋n ⋉ ℍn

Let [Formula: see text] be the [Formula: see text]-dimensional Heisenberg group and let [Formula: see text] be the [Formula: see text]-dimensional torus acting on [Formula: see text] by automorphisms. We consider the semidirect product group [Formula: see text] The cortex, [Formula: see text] of [Formula: see text] is the set of all unitary irreducible representations [Formula: see text] in the unitary dual [Formula: see text] of [Formula: see text] that cannot be Hausdorff separated from the identity representation [Formula: see text] of [Formula: see text] In this paper, we describe explicitly the cortex ([Formula: see text]) of [Formula: see text] using the coadjoint orbits of the group.

R-Harmonic and Complex Isoparametric Functions on the Lie Groups {{mathbb {R}}}^m ltimes {{mathbb {R}}}^n and {{mathbb {R}}}^m ltimes mathrm {H}^{2n+1}

In this paper we introduce the notion of complex isoparametric functions on Riemannian manifolds. These are then employed to devise a general method for constructing proper r-harmonic functions. We then apply this to construct the first known explicit proper r-harmonic functions on the Lie group semidirect products {{mathbb {R}}}^m ltimes {{mathbb {R}}}^n and {{mathbb {R}}}^m ltimes mathrm {H}^{2n+1}, where mathrm {H}^{2n+1} denotes the classical (2n+1)-dimensional Heisenberg group. In particular, we construct such examples on all the simply connected irreducible four-dimensional Lie groups.

The Tangential $$\varvec{k}$$-Cauchy–Fueter Operator and $$\varvec{k}$$-CF Functions Over the Heisenberg Group

In this paper, we investigate quaternionic analysis on the $$(4n+1)$$-dimensional Heisenberg group. The tangential k-Cauchy–Fueter operator and k-CF functions are counterparts of the tangential Cauchy–Riemann operator and CR functions on the Heisenberg group in the theory of several complex variables, respectively. We give the Penrose integral formula for k-CF functions and establish the Bochner–Martinelli formula for the tangential k-Cauchy–Fueter operator. We also construct the tangential k-Cauchy–Fueter complex.

Laplacian coflow on the $7$-dimensional Heisenberg group

We study the Laplacian coflow and the modified Laplacian coflow of $G_2$-structures on the $7$-dimensional Heisenberg group. For the Laplacian coflow we show that the solution is always ancient, that is it is defined in some interval $(-\infty,T)$, with $0<T<+\infty$. However, for the modified Laplacian coflow, we prove that in some cases the solution is defined only on a finite interval while in other cases the solution is ancient or eternal, that is it is defined on $(-\infty, \infty)$.

Spherical-Radial Multipliers on the Heisenberg Group

Let Hn be the (2n+1)-dimensional Heisenberg group. We consider a radial Fourier multiplier which is a spherical function on Hn and show that it is a Herz-Schur multiplier.

Heat and entropy flows in Carnot groups

We prove the correspondence between the solutions of the sub-elliptic heat equation in a Carnot group \mathbb G and the gradient flows of the relative entropy functional in the Wasserstein space of probability measures on \mathbb G . Our result completely answers a question left open in a previous paper by N. Juillet, where the same correspondence was proved for \mathbb G = \mathbb H^n , the n -dimensional Heisenberg group.

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.

Corwin–Greenleaf multiplicity function for compact extensions of the Heisenberg group

Let [Formula: see text] be the [Formula: see text]-dimensional Heisenberg group and [Formula: see text] a closed subgroup of [Formula: see text] acting on [Formula: see text] by automorphisms such that [Formula: see text] is a Gelfand pair. Let [Formula: see text] be the semidirect product of [Formula: see text] and [Formula: see text]. Let [Formula: see text] be the respective Lie algebras of [Formula: see text] and [Formula: see text], and [Formula: see text] the natural projection. For coadjoint orbits [Formula: see text] and [Formula: see text], we denote by [Formula: see text] the number of [Formula: see text]-orbits in [Formula: see text], which is called the Corwin–Greenleaf multiplicity function. In this paper, we give two sufficient conditions on [Formula: see text] in order that [Formula: see text] For [Formula: see text], assuming furthermore that [Formula: see text] and [Formula: see text] are admissible and denoting respectively by [Formula: see text] and [Formula: see text] their corresponding irreducible unitary representations, we also discuss the relationship between [Formula: see text] and the multiplicity [Formula: see text] of [Formula: see text] in the restriction of [Formula: see text] to [Formula: see text]. Especially, we study in Theorem 4 the case where [Formula: see text]. This inequality is interesting because we expect the equality as the naming of the Corwin–Greenleaf multiplicity function suggests.

Concentration-compactness principle for Trudinger–Moser inequalities on Heisenberg groups and existence of ground state solutions

Let $\mathbb{H}^{n}=\mathbb{C}^{n}\times\mathbb{R}$ be the $n$-dimensional Heisenberg group, $Q=2n+2$ be the homogeneous dimension of $\mathbb{H}^{n}$. We extend the well-known concentration-compactness principle on finite domains in the Euclidean spaces of \ P. L. Lions to the setting of the Heisenberg group $\mathbb{H}^{n}$. Furthermore, we also obtain the corresponding concentration-compactness principle for the Sobolev space $HW^{1,Q}\left( \mathbb{H}^{n}\right) $ on the entire Heisenberg group $\mathbb{H}^{n}$. Our results improve the sharp Trudinger-Moser inequality on domains of finite measure in $\mathbb{H}^{n}$ by Cohn and the second author [8] and the corresponding one on the whole space $\mathbb{H}^n$ by Lam and the second author [21]. All the proofs of the concentration-compactness principles in the literature even in the Euclidean spaces use the rearrangement argument and the Poly\'a-Szeg\"{o} inequality. Due to the absence of the Poly\'a-Szeg\"{o} inequality on the Heisenberg group, we will develop a different argument. Our approach is surprisingly simple and general and can be easily applied to other settings where symmetrization argument does not work. As an application of the concentration-compactness principle, we establish the existence of ground state solutions for a class of $Q$- Laplacian subelliptic equations on $\mathbb{H}^{n}$ with nonlinear terms $f$ of maximal exponential growth $\exp\left( \alpha t^{\frac{Q}{Q-1}}\right) $ as $t\rightarrow+\infty$.

CMC A-net surfaces in three dimensional Heisenberg group

CMC A-net surfaces in three dimensional Heisenberg group