Carnot-Caratheodory Spaces Research Articles

Privileged Coordinates for Carnot–Carathéodory Spaces of Lower Smoothness

We describe classes of local coordinates on the Carnot–Caratheodory spaces of lower smoothness which permit the homogeneous approximation of quasimetrics and basis vector fields. We establish the minimal smoothness that is required for these classes to coincide with the class of the already-described privileged coordinates in the infinite smoothness case. Moreover, we apply these results to prove the analogs of the available theorems in the case of the canonical coordinates of the second kind. Also, we prove some convergence theorems in quasimetric spaces.

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

The notion of a sub-Lorentzian structure with multidimensional time on Carnot–Caratheodory spaces is introduced. It is proved that classes of graph surfaces are spacelike, and a sub-Lorentzian analogue of the area formula is proved.

On Local Metric Characteristics of Level Sets of C1H-Mappings of Carnot Manifolds

Considering the level surfaces of the mappings of class C1 which are defined on Carnot manifolds and take values in Carnot—Caratheodory spaces, we introduce some adequate local metric characteristic that bases on a correspondence with a neighborhood of the kernel of the sub-Riemannian differential. Moreover, for the mappings on Carnot groups we construct an adapted basis in the preimage which matches local sub-Riemannian structures on the complement of the kernel of the sub-Riemannian differential (including those meeting the level set) and on the arrival set.

On the Class of Hölder Surfaces in Carnot—Carathéodory Spaces

Considering two classes of Holder mappings of Carnot-Caratheodory 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.

Local metric properties of level surfaces on Carnot–Caratheodory spaces

For level sets of -mappings of Carnot manifolds to Carnot–Caratheodory spaces, we introduce an adequate local metric characteristic. Moreover, for mappings defined on Carnot groups, we construct a special adapted basis in the preimage such that it assigns a suitable local sub-Riemannian structure on a complement of a kernel of a sub-Riemannian differential to the initial sub-Riemannian structure in the image.

Polynomial Sub-Riemannian Differentiability on Carnot–Carathéodory Spaces

We show that the classes of Holder mappings of Carnot–Caratheodory 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.

The Local Approximation Theorem in Various Coordinate Systems

We find some sufficient conditions on the local coordinate system of a Carnot–Caratheodory space of low smoothness that ensure Gromov-type estimates on the divergence of local (quasi)metrics. We also obtain these estimates for the canonical coordinate system of the second kind and various mixed coordinate systems.

Dirichlet and Neumann eigenvalue problems on CR manifolds

We study the properties of Carnot–Caratheodory spaces attached to a strictly pseudoconvex CR manifold M, in a neighborhood of each point $$x \in M$$ , versus the pseudohermitian geometry of M arising from a fixed positively oriented contact form $$\theta $$ on M. The weak Dirichlet problem for the sublaplacian $$\Delta _b$$ on $$(M, \theta )$$ is solved on domains $$\Omega \subset M$$ supporting the Poincare inequality. The solution to Neumann problem for the sublaplacian $$\Delta _b$$ on a $$C^{1,1}$$ connected $$(\epsilon , \delta )$$ -domain $$\Omega \subset {{\mathbb {G}}}$$ in a Carnot group (due to Danielli et al. in: Memoirs of American Mathematical Society 2006) is revisited for domains in a CR manifold. As an application we prove discreetness of the Dirichlet and Neumann spectra of $$\Delta _b$$ on $$\Omega \subset M$$ in a Carnot–Cartheodory complete pseudohermitian manifold $$(M, \theta )$$ .

The Isoperimetric Problem in Carnot-Caratéodory Spaces

We present some recent results obtained on the isoperimetric problem in a class of Carnot-Caratheodory spaces, related to the Heisenberg group. This is the framework of Pansu’s conjecture about the shape of isoperimetric sets. Two different approaches are considered. On one hand we describe the isoperimetric problem in Grushin spaces, under a symmetry assumption that depends on the dimension and we provide a classification of isoperimetric sets for special dimensions. On the other hand, we present some results about the isoperimetric problem in a family of Riemannian manifolds approximating the Heisenberg group. In this context we study constant mean curvature surfaces. Inspired by Abresch and Rosenberg techniques on holomorphic quadratic differentials, we classify isoperimetric sets under a topological assumption.

Existence of tangent lines to Carnot–Carathéodory geodesics

We show that length minimizing curves in Carnot–Caratheodory 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.

Area of Some Hölder Surfaces on Carnot–Carathéodory Spaces

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

On an analogue of parallel translation on Carnot–Carathéodory spaces

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

Approximation of Hölder mappings on Carnot–Carathéodory spaces

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

Variational mean curvatures in abstract measure spaces

Variational mean curvatures were introduced by Massari (Arch Ration Mech Anal 55:357–382, 1974) in the Euclidean setting. Later, Barozzi et al. (Proc AMS 99:313–316, 1987) proved that any set of finite perimeter has a summable variational mean curvature. This result was strongly improved by Barozzi (Rend Mat Acc Lincei S 9(5):149–159, 1994). The main goal of this paper is to show that the construction of variational mean curvatures can be still performed in an abstract measure setting, by axiomatizing a few abstract properties that any perimeter should have. This includes perimeters in suitable metric measure spaces, as introduced in Miranda in (J Math Pure Appl 82:975–1004, 2003) (which includes a variety of situations, as for example, Carnot-Caratheodory spaces), perimeters in spaces with density and nonlocal or fractional perimeters as well.

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

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

Graph surfaces over three-dimensional Carnot–Carathéodory spaces

The polynomial sub-Riemannian differentiability of the graph mappings of functions defined on three-dimensional Carnot–Caratheodory spaces is proved and an area formula for graph surfaces is derived.

Embedding theorem on RD-spaces

An RD-space $(X, d,\mu)$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds. An important class of RD-spaces is provided by Carnot-Caratheodory spaces with a doubling measure. In this article, the author establishes the embedding theorem for Besov and Triebel-Lizorkin spaces on RD-spaces.

One-Dimensional Level Sets of hc-Differentiable Mappings of Carnot–Carathéodory Spaces

We study continuously hc-differentiable mappings from a Carnot-Caratheodory space M such that dim HgM =d imTgM − 1= N for every g ∈ M to an Euclidean N - dimensional space with the property that the hc-differential of the mapping is surjective. We prove that the level set of such a mapping is a curve with Hausdorff dimension 2 in the sub-Riemannian metric. We obtain area formulas for curves of that kind. Bibliography :1 7titles. In this paper, we study the structure and metric properties of level sets of hc-differentiable mappings from a Carnot-Caratheodory space to the Euclidean space. We consider an equiregular Carnot-Caratheodory space (1) that is a connected smooth Riemannian manifold M in the tangent bundle TM with a distinguished subbundle HM (called horizontal) satisfying the Hormander condition (2). By this condition, any two points of M can be joined by ah orizontal curve, i.e., a piecewise smooth curve such that its tangent vector at each point belongs to the horizontal subbundle HM .M oreover, theCarnot-Caratheodory metric dc(x, y), x, y ∈ M ,i s naturally defined as the infimum of lengths of all horizontal curves joining x and y.

The Poincaré inequality for C 1,α-smooth vector fields

We obtain the Poincare inequality for the equiregular Carnot-Caratheodory spaces spanned by vector fields with Holder class derivatives.

Fine properties of basis vector fields on Carnot-Carathéodory spaces under minimal assumptions on smoothness

We deduce some fine geometric properties of basis vector fields on Carnot-Caratheodory spaces under minimal assumptions on smoothness. These lead to the estimates of the approximation of a Carnot-Caratheodory space by local homogeneous groups.