Finding Einstein solvmanifolds by a variational method

We use a variational approach to prove that any nilpotent Lie algebra having a codimension-one abelian ideal, and anyone of dimension , admits a rank-one solvable extension which can be endowed with an Einstein left-invariant riemannian metric. A curve of -dimensional Einstein solvmanifolds is also given. Received: 29 May 2001; in final form: 4 October 2001 / Published online: 4 April 2002     

On invariant volumes of codimension-one Anosov flows and the Verjovsky conjecture

We show that any topologically transitive codimension-one Anosov flow on a closed manifold is topologically equivalent to a smooth Anosov flow that preserves a smooth volume. By a classical theorem due to Verjovsky, any higher-dimensional codimension-one Anosov flow is topologically transitive. Recently, Simić showed that any higher-dimensional codimension-one Anosov flow that preserves a smooth volume is topologically equivalent to the suspension of an Anosov diffeomorphism. Therefore, our result gives a complete classification of codimension-one Anosov flows up to topological equivalence in higher dimensions.     

Hausdorff dimension distribution of quasiconformal mappings on the Heisenberg group

We construct quasiconformal mappings on the Heisenberg group which change the Hausdorff dimension of Cantor-type sets in an arbitrary fashion. On the other hand, we give examples of subsets of the Heisenberg group whose Hausdorff dimension cannot be lowered by any quasiconformal mapping. For a general set of a certain Hausdorff dimension we obtain estimates of the Hausdorff dimension of the image set in terms of the magnitude of the quasiconformal distortion.     

Geodesics in Heisenberg Groups

We obtain equations of geodesic lines in Heisenberg groups H²ⁿ⁺¹and prove that the ideal boundary of the Heisenberg group H²ⁿ⁺¹is a sphere S^2n-1with a natural CR-structure and corresponding Carnot-Carathéodory metric, i.e. it is a one-point compactification of the Heisenberg group H^2n-1of the next dimension in a row.     

Cohomology and deformations for the Heisenberg Hom-Lie algebras

ABSTRACT

In this work, we consider the Heisenberg Lie algebra with all its Hom-Lie structures. We completely characterize the cohomology and deformations of any order of all Heisenberg Hom-Lie algebras of dimension 3.     

Differentiability of Intrinsic Lipschitz Functions within Heisenberg Groups

We study the notion of intrinsic Lipschitz graphs within Heisenberg groups, focusing our attention on their Hausdorff dimension and on the almost everywhere existence of (geometrically defined) tangent subgroups. In particular, a Rademacher type theorem enables us to prove that previous definitions of rectifiable sets in Heisenberg groups are natural ones.     

Splittings of finitely generated groups over two-ended subgroups

We describe a means of constructing splittings of a one-ended finitely generated group over two-ended subgroups, starting with a finite collection of codimension-one two-ended subgroups. In the case where all the two-ended subgroups have two-ended commensurators, we obtain an annulus theorem, and a form of the JSJ splitting of Rips and Sela. The construction uses ideas from the work of Dunwoody, Sageev and Swenson. We use a particular kind of order structure which combines cyclic orders and treelike structures. In the special case of hyperbolic groups, this provides a link between combinarorial constructions, and constructions arising from the topological structure of the boundary. In this context, we recover the annulus theorem of Scott and Swarup. We also show that a one-ended finitely generated group which contains an infinite-order element, and such that every infinite cyclic subgroup is (virtually) codimension-one is a virtual surface group.

     

Horizontal submanifolds of groups of Heisenberg type

We study maximal horizontal subgroups of Carnot groups of Heisenberg type. We classify those of dimension half of that of the canonical distribution (“lagrangians”) and illustrate some notable ones of small dimension. An infinitesimal classification of the arbitrary maximal horizontal submanifolds follows as a consequence. This work was supported by CONICET, Antorchas, FONCyT and Secyt (UNC).     

一类特殊的Heisenberg3-李代数的结构

本文研究一类特殊的Heisenberg 3-李代数的结构.给出其内导子代数与导子代数的具体表示形式,并证明了两种同维数的Heisenberg 3-李代数是不同构的.     

Best Constants for Moser-Trudinger Inequalities, Fundamental Solutions and One-Parameter Representation Formulas on Groups of Heisenberg Type

We derive the explicit fundamental solutions for a class of degenerate (or singular) one-parameter subelliptic differential operators on groups of Heisenberg (H) type. This extends the results of Kaplan of the sub-Laplacian on H-type groups, which in turn generalizes Folland's result on the Heisenberg group. As an application, we obtain a one-parameter representation formula for Sobolev functions of compact support on H-type groups. By choosing the parameter equal to the homogeneous dimension Q and using the Moser-Trudinger inequality for the convolutional type operator on stratified groups obtained in [18], we get the following theorem which gives the best constant for the Moser-Trudinger inequality for Sobolev functions in H-type groups. Let ${\Bbb G}We derive the explicit fundamental solutions for a class of degenerate (or singular) one-parameter subelliptic differential operators on groups of Heisenberg (H) type. This extends the results of Kaplan of the sub-Laplacian on H-type groups, which in turn generalizes Folland's result on the Heisenberg group. As an application, we obtain a one-parameter representation formula for Sobolev functions of compact support on H-type groups. By choosing the parameter equal to the homogeneous dimension Q and using the Moser-Trudinger inequality for the convolutional type operator on stratified groups obtained in [18], we get the following theorem which gives the best constant for the Moser-Trudinger inequality for Sobolev functions in H-type groups. Let ? be any group of Heisenberg type whose Lie algebra is g enerated by m left invariant vector fields and with a q-dimensional center. Let and Then, with A _Q as the sharp constant, where ∇_? denotes the subellitpic gradient on ? This continues the research originated in our earlier study of the best constants in Moser-Trudinger inequalities and fundamental solutions for one-parameter subelliptic operators on the Heisenberg group [18]. Received March 15, 2001, Accepted September 21, 2001     

Extrinsic Geometric Flows on Codimension-One Foliations

We study the geometry of a codimension-one foliation with a time-dependent Riemannian metric. The work begins with formulae for deformations of geometric quantities as the Riemannian metric varies along the leaves of a foliation. Then the Extrinsic Geometric Flow depending on the second fundamental form of the foliation is introduced. Under suitable assumptions, this evolution yields the second-order parabolic PDEs, for which the existence/uniqueness and in some cases convergence of a solution are shown. Applications to the problem of prescribing the mean curvature function of a codimension-one foliation, and examples with harmonic and umbilical foliations (e.g., foliated surfaces) and with twisted product metrics are given.     

On simple modules over conformal Galilei algebras

We study irreducible representations of two classes of conformal Galilei algebras in 1-spatial dimension. We construct a functor which transforms simple modules with nonzero central charge over the Heisenberg subalgebra into simple modules over the conformal Galilei algebras. This can be viewed as an analogue of oscillator representations. We use oscillator representations to describe the structure of simple highest weight modules over conformal Galilei algebras. We classify simple weight modules with finite dimensional weight spaces over finite dimensional Heisenberg algebras and use this classification and properties of oscillator representations to classify simple weight modules with finite dimensional weight spaces over conformal Galilei algebras.     

Quasiconvexity in the Heisenberg group

We show that if A is a closed subset of the Heisenberg group whose vertical projections are nowhere dense, then the complement of A is quasiconvex. In particular, closed sets which are null sets for the cc-Hausdorff 3-measure have quasiconvex complements. Conversely, we exhibit a compact totally disconnected set of Hausdorff dimension three whose complement is not quasiconvex.     

Symplectic structures¶on Heisenberg-type nilmanifolds

We use cohomological methods to study the existence of symplectic structures on nilmanifolds associated to two-step nilpotent Lie groups. We construct a new family of symplectic nilmanifolds with building blocks the quaternionic analogue of the Heisenberg group, determining the dimension of the space of all left invariant symplectic structures. Such structures can not be K?hlerian. Also, we prove that the nilmanifolds associated to H type groups are not symplectic unless they correspond to the classical Heisenberg groups. Received: 26 May 1999 / Revised version: 10 April 2000     

Entire parabolic trajectories as minimal phase transitions

For the class of anisotropic Kepler problems in $\mathbb{R }^d\setminus \{0\}$ with homogeneous potentials, we seek parabolic trajectories having prescribed asymptotic directions at infinity and which, in addition, are Morse minimizing geodesics for the Jacobi metric. Such trajectories correspond to saddle heteroclinics on the collision manifold, are structurally unstable and appear only for a codimension-one submanifold of such potentials. We give them a variational characterization in terms of the behavior of the parameter-free minimizers of an associated obstacle problem. We then give a full characterization of such a codimension-one manifold of potentials and we show how to parameterize it with respect to the degree of homogeneity.     

Differential Algebra and Liouvillian first integrals of foliations

Intuitively, a complex Liouvillian function is one that is obtained from complex rational functions by a finite process of integrations, exponentiations and algebraic operations. In the framework of ordinary differential equations the study of equations admitting Liouvillian solutions is related to the study of ordinary differential equations that can be integrated by the use of elementary functions, that is, functions appearing in the Differential Calculus. A more precise and geometrical approach to this problem naturally leads us to consider the theory of foliations. This paper is devoted to the study of foliations that admit a Liouvillian first integral. We study holomorphic foliations (of dimension or codimension one) that admit a Liouvillian first integral. We extend results of Singer (1992) [20] related to Camacho and Scárdua (2001) [4], to foliations on compact manifolds, Stein manifolds, codimension-one projective foliations and germs of foliations as well.     

Complete signal processing bases and the Jacobi group

The continuous windowed Fourier and wavelet transforms are created from the actions of the Heisenberg and affine groups, respectively. Both wavelet and windowed Fourier bases are known to be complete; that is, the only signal which is orthogonal to every element of each basis is the zero signal. The Jacobi group is a group which contains both the Heisenberg and affine groups, and it can also be used to produce bases for signal processing. This paper investigates completeness for bases of one and two real variables which are produced by the Jacobi group.     

Some ergodic and rigidity properties of discrete Heisenberg group actions

The goal of this paper is to study ergodic and rigidity properties of smooth actions of the discrete Heisenberg group \(\mathcal{H}\). We establish the decomposition of the tangent space of any C^∞ compact Riemannian manifold M for Lyapunov exponents, and show that all Lyapunov exponents for the central elements are zero. We obtain that if an \(\mathcal{H}\) action contains an Anosov element, then under certain conditions on the eigenvalues of this element, the action of each central element is of finite order. In particular, there is no faithful codimension one Anosov Heisenberg group action on any compact manifold, and there is no faithful codimension two Anosov Heisenberg group action on tori. In addition, we show smooth local rigidity for higher rank ergodic \(\mathcal{H}\) actions by toral automorphisms, using a generalization of the KAM (Kolmogorov–Arnold–Moser) iterative scheme.     

Algebraic actions of the discrete Heisenberg group: Expansiveness and homoclinic points

We survey some of the known criteria for expansiveness of principal algebraic actions of countably infinite discrete groups. In the special case of the discrete Heisenberg group we propose a new approach to this problem based on Allan’s local principle.     

Contact quasiconformal immersions

Contact immersions of contact manifolds endowed with the associated Carnot-Carathéodory (CC) metric (for example, immersions of the Heisenberg group H ³ ～ ? _CC ³ in itself) are considered. It is assumed that the manifolds have the same dimension and the immersions are quasiconformal with respect to the CC metric. The main assertion is as follows: A quasiconformal immersion of the Heisenberg group in itself, just as a quasiconformal immersion of any contact manifold of conformally parabolic type in a simply connected contact manifold, is globally injective; i.e., such an immersion is an embedding, which, in addition, is surjective in the case of the Heisenberg group. Thus, the global homeomorphism theorem, which is well known in the space theory of quasiconformal mappings, also holds in the contact case.