Metric Properties of Graphs on Carnot–Carathéodory Spaces with Sub-Lorentzian Structure

Doklady Mathematics - The notion of a sub-Lorentzian structure with multidimensional time on Carnot–Carathéodory spaces is introduced. It is proved that classes of graph surfaces are...     

Three-Dimensional Graph Surfaces on Five-Dimensional Carnot–Carathéodory Spaces

We study the class of codimension 2 graph surfaces over some three-dimensional Lie groups and establish some analogs of the differential properties of the corresponding graph mappings. Moreover, we derive the area formula and describe the classes of minimal surfaces of codimension 1 and 2.     

Area of Some Hölder Surfaces on Carnot–Carathéodory Spaces

An area formula for classes of Hölder mappings on Carnot–Carathéodory spaces is proved. As important particular cases, general and classical versions of surface measure calculation for graph surfaces are considered.     

Polynomial Sub-Riemannian Differentiability on Carnot–Carathéodory Spaces

We show that the classes of Hölder mappings of Carnot–Carathéodory spaces are polynomially differentiable in the sub-Riemannian sense. Moreover, we prove the existence of intrinsic (or adapted) bases, which enable us to match the nonholonomic structures of the images of mappings and target spaces.     

On the Class of Hölder Surfaces in Carnot—Carathéodory Spaces

Considering two classes of Hölder mappings of Carnot-Carathéodory spaces, graph mappings and smooth mappings in the Riemannian sense, we obtain an area formula for image surfaces. In particular, we describe the structure of the polynomial sub-Riemannian differential for graph mappings.

     

A Coarea Formula for Smooth Contact Mappings of Carnot–Carathéodory Spaces

We prove the coarea formula for sufficiently smooth contact mappings of Carnot manifolds to Carnot–Carathéodory spaces. In particular, we investigate level surfaces of these mappings, and compare Riemannian and sub-Riemannian measures on them. Our main tool is the sharp asymptotic behavior of the Riemannian measure of the intersection of a tangent plane to a level surface and a sub-Riemannian ball. This calculation in particular implies that the sub-Riemannian measure of the set of characteristic points (i.e., the points at which the sub-Riemannian differential is degenerate) equals zero on almost every level set.     

Metric Properties of Level Surfaces of Hölder Mappings Defined on Two-Step Carnot Groups

For a model case of Hölder mappings of two-step nilpotent graded groups, the metric properties of level surfaces are established. A version of the coarea formula for noncontact mappings is deduced as an application.     

Approximation of Hölder mappings on Carnot–Carathéodory spaces

The general nature of approximation of certain Hölder mappings of Carnot–Carathéodory spaces is described. The local existence of a basis in the image which preserves sub-Riemannian Hausdorff dimension is also proved.     

On an analogue of parallel translation on Carnot–Carathéodory spaces

A new idea of constructing the parallel translation map on a Carnot–Carathéodory space is proposed, the basic properties of this map are described, and the polynomials approximating graph mappings are qualitatively described.     

Existence of tangent lines to Carnot–Carathéodory geodesics

We show that length minimizing curves in Carnot–Carathéodory spaces possess at any point at least one tangent curve (i.e., a blow-up in the nilpotent approximation) equal to a straight horizontal line.     

Hardy–Sobolev Type Inequalities with Sharp Constants in Carnot–Carathéodory Spaces

We prove a generalization with sharp constants of a classical inequality due to Hardy to Carnot groups of arbitrary step, or more general Carnot–Carathéodory spaces associated with a system of vector fields of Hörmander type. Under a suitable additional assumption (see Eq. 1.6 below) we are able to extend such result to the nonlinear case \(p\not= 2\). We also obtain a sharp inequality of Hardy–Sobolev type.     

Graph surfaces of codimension two over three-dimensional Carnot–Carathéodory spaces

An area formula for graph surfaces of codimension two over three-dimensional Carnot–Carathéodory spaces is derived and applied to obtain basic properties of minimal surfaces.     

Analytic Description of the Spaces Dual to Spaces of Holomorphic Functions of Given Growth on Carathéodory Domains

The spaces dual to spaces of holomorphic functions of given growth on Carathéodory domains are described by using the Cauchy transform of functionals. A pseudoanalytic extension of such transforms to the whole plane is constructed, which makes it possible to remove convexity constrains and consider spaces determined by weights of general form, rather than only by those whose dependence on the distance from a point of the domain to its boundary is one-dimensional.     

A remark on Carathéodory balls

Member of URA CNRS D 1322 Groupes de Lie et Géométrie     

Carnot–Carathéodory metrics in unbounded subdomains of {{\mathbb{C}}^2}

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 the real and imaginary parts of the CR vector field.     

Non-Commutative Carathéodory Interpolation

We prove a Carathéodory–Fejér type interpolation theorem for certain matrix convex sets in mathbbC^d{mathbb{C}^d} using the Blecher–Ruan–Sinclair characterization of abstract operator algebras. Our results generalize the work of Dmitry S. Kalyuzhnyĭ-Verbovetzkiĭ for the d-dimensional non-commutative polydisc.     

On vector lattices of elementary Carathéodory functions

In this paper we deal with the vector lattice C(B) of all elementary Carathéodory functions corresponding to a generalized Boolean algebra B.This work was supported by grant VEGA 2/1131/21.     

Borel–Carathéodory Type Theorem for Monogenic Functions

In this paper we give a generalization of the classical Borel–Carathéodory theorem in complex analysis to higher dimensions in the framework of Quaternionic Analysis. Submitted: November 14, 2007., Revised: February 25, 2008.