Related papers: Geometry of $1$-codimensional measures in Heisenbe…
This paper is devoted to show that the flatness of tangents of $1$-codimensional measures in Carnot Groups implies $C^1_\mathbb{G}$-rectifiability. As applications we prove that measures with $(2n+1)$-density in the Heisenberg groups…
In this paper the classification of left-invariant Riemannian metrics, up to the action of the automorphism group, on cotangent bundle of (2n+1)-dimensional Heisenberg group is presented. Also, it is proved that the complex structure on…
We initiate a classification of uniform measures in the first Heisenberg group $\mathbb H$ equipped with the Kor\'anyi metric $d_H$, that represents the first example of a noncommutative stratified group equipped with a homogeneous…
A general approach to compute the spherical measure of submanifolds in homogeneous groups is provided. We focus our attention on the homogeneous tangent space, that is a suitable weighted algebraic expansion of the submanifold. This space…
In this paper we aim at identifying the level sets of the gauge norm in the Heisenberg group $\mathbb{H}^n$ via the prescription of their (non-constant) horizontal mean curvature. We establish a uniqueness result in $\mathbb{H}^1$ under an…
We study points of density 1/2 of sets of finite perimeter in infinite-dimensional Gaussian spaces and prove that, as in the finite-dimensional theory, the surface measure is concentrated on this class of points. Here density 1/2 is…
Various problems concerning the geometry of the space $u^*(\cH)$ of Hermitian operators on a Hilbert space $\cH$ are addressed. In particular, we study the canonical Poisson and Riemann-Jordan tensors and the corresponding foliations into…
In this paper, we study some basic geometric properties of pseudohermitian submanifolds of the Heisenberg groups. In particular, we obtain the uniqueness and existence theorems, and some rigidity theorems.
We prove sharp geometric rigidity estimates for isometries on Heisenberg groups. Our main result asserts that every $(1+\varepsilon)$-quasi-isometry on a John domain of the Heisenberg group $\mathbb{H}^n$, $n>1$, is close to some isometry…
We prove quantitative stability of isometries on the first Heisenberg group with sub-Riemannian geometry: every $ (1+ \varepsilon)$-quasi-isometry of the John domain of the Heisenberg group $ \mathbb {H} $ is close to some isometry with…
We extend Nguyen's characterization of Sobolev spaces $W^{1,p}$ to the setting of PI-metric measure spaces such that at a.e. point the tangent space (in the Gromov-Hausdorff sense) is unique and Euclidean with a fixed dimension. We also…
We investigate metric properties of level sets of horizontally differentiable maps defined on the first Heisenberg group $(\Bbb{H}^1,d_{cc})$ equipped with the standard sub-Riemannian structure. In particular, we present an exhaustive…
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 $\mathbb{C}^n$. The generic topological structure of…
This paper aims to show that there exists a triangulation of the Heisenberg group $\mathbb{H}^n$ into singular simplexes with regularity properties on both the low-dimensional and high-dimensional layers. For low dimensions, we request our…
Supersymmetric domain-wall spacetimes that lift to Ricci-flat solutions of M-theory admit generalized Heisenberg (2-step nilpotent) isometry groups. These metrics may be obtained from known cohomogeneity one metrics of special holonomy by…
We prove some geometric properties of sets in the first Heisenberg group whose Heisenberg Hausdorff dimension is the minimal or maximal possible in relation to their Euclidean one and the corresponding Hausdorff measures are positive and…
We are interested in studying doubling metric spaces with the property that at some of the points the metric tangent is unique. In such a setting, Finsler-Carnot-Caratheodory geometries and Carnot groups appear as models for the tangents.…
This paper establishes connections between the group-Fourier transform and the geometry of measures in the Heisenberg group. Firstly, it is shown that if the Fourier transform of a compactly supported, finite, Radon measure is square…
A natural notion of higher order rectifiability is introduced for subsets of Heisenberg groups $\mathbb{H}^n$ in terms of covering a set almost everywhere by a countable union of $(\mathbf{C}_H^{1,\alpha},\mathbb{H})$-regular surfaces, for…
The notion of $\Gamma$-symmetric space is a natural generalization of the classical notion of symmetric space based on $\Z_2$-grading of Lie algebras. In our case, we consider homogeneous spaces $G/H$ such that the Lie algebra $\g$ of $G$…