Counter example to the Besicovitch covering property for some Carnot groups equipped with their Carnot-Carathéodory metric

We construct a counter example to the Besicovitch covering property for some class of Carnot groups equipped with their Carnot - Carathéodory metric. The construction uses some regularity assumptions for minimal geodesics and for the distance function to the origin. As an example we show that groups of type H, thus including all Heisenberg groups, satisfy these assumptions. We prove in particular that, in groups of type H, the distance function to the origin is Pansu-differentiable outside the center with continuous Pansu-differential there.Mathematics Subject Classification (2000): 53C17 (22E25)     

A nonholonomic Laplace operator

In this paper first the Laplace operator on a completely nonholonomic Riemannian manifold is defined in an invariant manner and its properties are considered. The method presented for studying it, as well as for the study of other hypoelliptic operators, involves the use of the geometry of nonholonomic manifolds. The nonholonomic metric (Carnot-Carathéodory metric), the Carathéodory measure, and hypoharmonic functions are defined. A theorem on the comparison of the spectra is proved and the connection is established between the bases of eigenfunctions of the ordinary and nonholonomic Laplacians. Conjectures are formulated on the principal term of the spectral asymptotic expansion of the nonholonomic Laplacian, on the structure of the wave fronts, and on the propagation of singularities.Translated from Problemy Matematicheskogo Analiza, No. 11, pp. 96–108, 1990.     

The Heston Riemannian distance function

The Heston model is a popular stock price model with stochastic volatility that has found numerous applications in practice. In the present paper, we study the Riemannian distance function associated with the Heston model and obtain explicit formulas for this function using geometrical and analytical methods. Geometrical approach is based on the study of the Heston geodesics, while the analytical approach exploits the links between the Heston distance function and the Carnot–Carathéodory distance function in the Grushin plane. For the Grushin plane, we establish an explicit formula for the Legendre–Fenchel transform of the limiting cumulant generating function and prove a partial large deviation principle that is true only inside a special set.     

Estimates of the Carathéodory metric on the symmetrized polydisc

Estimates for the Carathéodory metric on the symmetrized polydisc are obtained. It is also shown that the Carathéodory and Kobayashi distances of the symmetrized three-disc do not coincide.     

Representations of Central Matrix-Valued Carathéodory Functions in Both Nondegenerate and Degenerate Cases

We derive left and right quotient representations for central q × q matrix-valued Carathéodory functions. Moreover, we obtain recurrent formulas for the matrix polynomials involved in the quotient representations. These formulas are the starting point for getting recurrent formulas for those matrix polynomials which occur in the Arov-Krein resolvent matrix for the nondegenerate matricial Carathéodory problem.     

On rank invariance of generalized Schwarz-Pick-Potapov block matrices of matrix-valued Carathéodory functions

In view of a multiple Nevanlinna-Pick interpolation problem, we study the rank of generalized Schwarz-Pick-Potapov block matrices of matrix-valued Carathéodory functions. Those matrices are determined by the values of a Carathéodory function and the values of its derivatives up to a certain order. We derive statements on rank invariance of such generalized Schwarz-Pick-Potapov block matrices. These results are applied to describe the case of exactly one solution for the finite multiple Nevanlinna-Pick interpolation problem and to discuss matrix-valued Carathéodory functions with the highest degree of degeneracy.     

Pseudodistances and Pseudometrics on real and complex manifolds

Summary In this paper we study the relationships between a class of distances and infinitesimal metrics on real and complex manifolds and their behavior under differentiable and holomorphic mappings. Some application to Riemannian and Finsler geometry are given and also new proofs and generalizations of some results of Royden, Harris and Reiffen on Kobayashi and Carathéodory metrics on complex manifolds are obtained. In particular we prove that on every complex manifold (finite or infinite- dimensional) the Kobayashi distance is the integrated form of the corresponding infinitesimal metric.     

Sub-Riemannian vs. Euclidean dimension comparison and fractal geometry on Carnot groups

We solve Gromov's dimension comparison problem for Hausdorff and box counting dimension on Carnot groups equipped with a Carnot-Carathéodory metric and an adapted Euclidean metric. The proofs use sharp covering theorems relating optimal mutual coverings of Euclidean and Carnot-Carathéodory balls, and elements of sub-Riemannian fractal geometry associated to horizontal self-similar iterated function systems on Carnot groups. Inspired by Falconer's work on almost sure dimensions of Euclidean self-affine fractals we show that Carnot-Carathéodory self-similar fractals are almost surely horizontal. As a consequence we obtain explicit dimension formulae for invariant sets of Euclidean iterated function systems of polynomial type. Jet space Carnot groups provide a rich source of examples.     

On 4-flattening theorems and the curves of Carathéodory, Barner and Segre

We discuss three classes of closed curves in the Euclidean space $\mathbb{R}^{3}$ which have non-vanishing curvature and at least 4 flattenings (points at which the torsion vanishes). Calling these classes (de.ned below) Barner, Segre and Carathéodory, we prove that Barner $\subset$ (Segre $\cap$ Carathéodory). We also prove that (Segre)\ (Segre $\cap$ Carathéodory) and (Carathéodory)\(Segre $\cap$ Carathéodory) are open sets in the space of closed smooth curves with the C-topology. Finally, we define a class of closed curves containing the class of Segre curves and -based on contact topology considerations, as the Huygens principle- we establish the conjecture that any curve of our class has at least 4 flattenings.     

Structure and the coefficients of certain classes of regular functions

In this paper, the well-known results of Carathéodory, Toeplitz, F. Riesz, Herglotz, containing the characterization of the Carathéodory class with the aid of the Stieltjes integral and with the aid of nonnegative quadratic forms (conditions on the coefficients), are extended to certain other classes of functions, regular in a circle or in an annulus.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 47–57, 1983.