arXiv research

Undergrad project: Shows geodesics coincide in Heisenberg group under two metrics.

The book explores Lie groups and Carnot-Carathéodory spaces, highlighting their applications in metric geometry and geometric group theory.

New insights into integrability and rectifiability in sub-Riemannian geometry.

The paper studies metrics and geodesics on a quaternionic Heisenberg group.

We introduce a new class of unbounded model subdomains of $\mathbb{C}^2$ for the $\Box_b$ problem. Unlike previous finite type models, these domains need not be bounded by algebraic varieties. In this paper we obtain precise global estimates for the Carnot-Carathéodory metric induced on the boundary of such domains by …

We consider the rate of volume growth of large Carnot-Carathéodory metric balls on a class of unbounded model hypersurfaces in $\mathbb{C}^2$. When the hypersurface has a uniform global structure, we show that a metric ball of radius $δ\gg 1$ either has volume on the order of $δ^3$ or $δ^4$. We also give necessary and …

In this paper we study the invariant Carnot-Caratheodory metrics on $SU(2)\simeq S^3$, $SO(3)$ and $SL(2)$ induced by their Cartan decomposition and by the Killing form. Beside computing explicitly geodesics and conjugate loci, we compute the cut loci (globally) and we give the expression of the Carnot-Caratheodory dis…

New definition of Rumin complex for nilpotent Lie groups.

We investigate the minimal surface problem in the three dimensional Heisenberg group, H, equipped with its standard Carnot-Caratheodory 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 prob…

We examine the theory of metric currents of Ambrosio and Kirchheim in the setting of spaces admitting differentiable structures in the sense of Cheeger and Keith. We prove that metric forms which vanish in the sense of Cheeger on a set must also vanish when paired with currents concentrated along that set. From this we…

Sobolev mappings preserve the Rumin complex on contact manifolds.

We study multi-parameter Carnot-Caratheodory balls, generalizing results due to Nagel, Stein, and Wainger in the single parameter setting. The main technical result is seen as a uniform version of the theorem of Frobenius. In addition, we study maximal functions associated to certain multi-parameter families of Carnot-…

Generalizes Cheeger inequality to Carnot-Carathéodory spaces.

An intrinsic definition in terms of conformal capacity is proposed for the conformal type of a Carnot--Carathéodory space (parabolic or hyperbolic). Geometric criteria of conformal type are presented. They are closely related to the asymptotic geometry of the space at infinity and expressed in terms of the isoperimetri…

We prove a quantitative bi-Lipschitz nonembedding theorem for the Heisenberg group with its Carnot-Carathéodory metric and apply it to give a lower bound on the integrality gap of the Goemans-Linial semidefinite relaxation of the Sparsest Cut problem.

A homogeneous Gibbons-Hawking ansatz is described, leading to 4-dimensional hyperkahler metrics with homotheties. In combination with Blaschke products on the unit disc in the complex plane, this ansatz allows one to construct infinite-dimensional families of such hyperkahler metrics that are, in a suitable sense, comp…

Study exact formula and geodesics for Carnot-Carathéodory distance on 2-step groups.

Extends scaling maps theory to manifolds with boundary.

Normal geodesic flows flows of Carnot-Caratheodory are discussed from the point of view of the theory of Hamiltonian systems. The geodesic flows corresponding to left-invariant metrics and left- and -right-invariant rank 2 distributions on the three-dimensional Heisenberg group are analysed as integrable systems. The f…

In this paper, we prove results concerning the large scale geometry of connected, simply connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics. Precisely, we prove that there do not exist quasi-isometric embeddings of such a nilpotent Lie group into either a CAT(0) metric space or an…

We find necessary and sufficient conditions for a Lipschitz map $f:\mathbb{R}E\to X$, into a metric space to have the image with the $k$-dimensional Hausdorff measure equal zero, $H^k(f(E))=0$. An interesting feature of our approach is that despite the fact that we are dealing with arbitrary metric spaces, we employ a …

We show that canonical Carnot-Caratheodory spherical and horospherical metrics, which are defined on the boundary at infinity of every rank one symmetric space of non-compact type, are visual, i.e., they are bilipschitz equivalent with universal bilipschitz constants to the inverse exponent of Gromov products based in …

The study constructs Yamabe operators on OC manifolds and proves their properties.

The paper examines convergence of distances in Lipschitz structures on manifolds.

We consider the sub-Riemannian metric $g_{h}$ on $\mathbb{S}^3$ provided by the restriction of the Riemannian metric of curvature 1 to the plane distribution orthogonal to the Hopf vector field. We compute the geodesics associated to the Carnot-Carathéodory distance and we show that, depending on their curvature, they …

We study the class of transversal submanifolds. We characterize their blow-ups at transversal points and prove a negligibility theorem for their "generalized characteristic set", with respect to the Carnot-Carathéodory Hausdorff measure. This set is made by all points of non-maximal degree. Observing that C^1 submanifo…

Carnot groups are distinguished spaces that are rich of structure: they are those Lie groups equipped with a path distance that is invariant by left-translations of the group and admit automorphisms that are dilations with respect to the distance. We present the basic theory of Carnot groups together with several remar…

The paper sketches a recent progress and formulates several open problems in studying equivariant quasiconformal and quasisymmetric homeomorphisms in negatively curved spaces as well as geometry and topology of noncompact geometrically finite negatively curved manifolds and their boundaries at infinity having Carnot--C…

The curvature discussed in this paper is a rather far going generalization of the Riemannian sectional curvature. We define it for a wide class of optimal control problems: a unified framework including geometric structures such as Riemannian, sub-Riemannian, Finsler and sub-Finsler structures; a special attention is p…

This paper shows that when the Riemannian metric on a contact manifold is blown up along the direction orthogonal to the contact distribution, the corresponding harmonic forms rescaled and normalized in the $L^2$-norms will converge to Rumin's harmonic forms. This proves a conjecture in Gromov `` Carnot-Caratheodory sp…

Gromov proposed to extract the (differential) geometric content of a sub-riemannian space exclusively from its Carnot-Carathéodory distance. One of the most striking features of a regular sub-riemannian space is that it has at any point a metric tangent space with the algebraic structure of a Carnot group, hence a homo…

We find a different approach to define convex functions in the sub-Riemannian setting. A function on a sub-Riemannian manifold is nonholonomically geodesic convex if its restriction to any nonholonomic (straightest) geodesic is convex. In the case of Carnot groups, this definition coincides with that by Danniell-Garofa…

This work is an investigation of perimeter measures in the metric measure space given by the Heisenberg group with Haar measure and a Carnot-Carathéodory metric, which is in general a sub-Finsler metric. Included is a reduction of Minkowski content in any CC-metric to an integral formula in terms of Lebesgue surface ar…

We consider Lie groups equipped with arbitrary distances. We only assume that the distance is left-invariant and induces the manifold topology. For brevity, we call such object metric Lie groups. Apart from Riemannian Lie groups, distinguished examples are sub-Riemannian Lie groups and, in particular, Carnot groups equ…

We introduce length dilatation structures on metric spaces, tempered dilatation structures and coherent projections and explore the relations between these objects and the Radon-Nikodym property and Gamma-convergence of length functionals. Then we show that the main properties of sub-riemannian spaces can be obtained f…

We deform a map into a Riemannian manifold that is horizontal with respect to a submersion onto a non-positively curved manifold and satisfies a Chow condition into a harmonic one through a horizontal homotopy.

The study examines optimal synthesis in a radially symmetric Grushin space with conditions on the weight function.

The paper proves existence and partial regularity for Legendrian area-minimizing currents.

For each submanifold of a stratified group, we find a number and a measure only depending on its tangent bundle, the grading and the fixed Riemannian metric. In two step stratified groups, we show that such number and measure coincide with the Hausdorff dimension and with the spherical Hausdorff measure of the submanif…

Let $M$ be a close complex manifold and $TM$ its holomorphic tangent bundle. We prove that if the global holomorphic sections of tangent bundle generate each fibre, then $M$ is a complex homogeneous manifold. Our proof depends on the complex version of Chow-Rashevskii theorem in Carnot-Caratheodory spaces.

We prove a quantitative openness theorem for $C^1$ submersions under suitable assumptions on the differential. We then apply our result to a class of exponential maps appearing in Carnot-Carathéodory spaces and we improve a classical completeness result by Palais.

In this paper we study the main geometric properties of the Carnot-Carathéodory (abbreviated CC) distance $\dc$ in the setting of $k$-step sub-Riemannian Carnot groups from many different points of view. An extensive study of the so-called normal CC-geodesics is given. We state and prove some related variational formul…

Here we study geodesics connecting two given points on odd-dimensional spheres respecting the Hopf fibration. This geodesic boundary value problem is completely solved in the case of 3-dimensional sphere and some partial results are obtained in the general case. The Carnot-Carathéodory distance is calculated. We also p…

We extend Bony's propagation of support argument \cite{Bony} to $C^1$ solutions of the non-homogeneous sub-elliptic $p-$Laplacian associated to a system of smooth vector fields satisfying Hörmander's finite rank condition. As a consequence we prove a strong maximum principle and strong comparison principle that general…

We characterize convex isoperimetric sets in the Heisenberg group endowed with horizontal perimeter. We first prove Sobolev regularity for a certain class of vector fields in the plane with bounded variation, related to the curvature equations. Then, by an approximation-reparameterization argument, we show that the bou…

Solves a long-standing problem on step-two groups with exact formulas.

We study the geometry associated to the Grusin operator G=Δ_{x}+|x|^{2}\partial_{u}^{2} on \mathbb{R}_{x}^{n}\times\mathbb{R}_{u}, to obtain heat kernel estimates for this operator. The main work is to find the shortest geodesics connecting two given points in $\mathbb{R}^{n+1}$. This gives the Carnot-Caratheodory dist…

We apply the theory of Weyl structures for parabolic geometries developed by A. Cap and J. Slovak in to compute, for a quaternionic contact (qc) structure, the Weyl connection associated to a choice of scale, i.e. to a choice of Carnot-Carathéodory metric in the conformal class. The result of this computation has appli…