A remark on Carathéodory balls

Member of URA CNRS D 1322 Groupes de Lie et Géométrie     

Extension of bounded holomorphic functions¶in convex domains

The E. Amar and G. Henkin theorem on the bounded extendability of bounded holomorphic functions from certain closed complex submanifolds of strictly pseudoconvex domains to the whole domain is generalized to the case of finite type convex domains and their intersections with affine linear hyperplanes. Suitable integral operators of Berndtsson–Andersson type are constructed and estimated for this purpose. Received: 7 July 2000     

Proper holomorphic self-maps of smooth bounded Reinhardt domains in C~n

It is proved that every proper holomorphic self-map of a smooth bounded Reinhardt domain of D'Angelo finite type in Cn (n > 1) is an automorphism.     

Szegö theorem,Carathéodory domains,and boundedness of calculating functionals

Convex Chebyshev sets M in a linear space X with norm or nonsymmetric norm ‖ · ‖ which are contained in a subspace H of X are considered. It is proved that if | { |_{H, φ} is the nonsymmetric norm on H determined by the Minkowski functional of , where B is the unit ball of X and ‖φ‖, with respect to 0, then M is a Chebyshev set in for any φ. From this result sufficient and necessary conditions for the convexity of Chebyshev sets and bounded Chebyshev sets contained in a subspace H of X are derived.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 3–15.Original Russian Text Copyright © 2005 by A. R. Alimov.     

Proper holomorphic self-maps of smooth bounded Reinhardt domains in C~2

It is proved that every proper holomorphic self-map of a smooth bounded Reinhardt domain in C~2 is an automorphism.     

The Carathéodory topology for multiply connected domains I

We consider the convergence of pointed multiply connected domains in the Carathéodory topology. Behaviour in the limit is largely determined by the properties of the simple closed hyperbolic geodesics which separate components of the complement. Of particular importance are those whose hyperbolic length is as short as possible which we call meridians of the domain. We prove continuity results on convergence of such geodesics for sequences of pointed hyperbolic domains which converge in the Carathéodory topology to another pointed hyperbolic domain. Using these we describe an equivalent condition to Carathéodory convergence which is formulated in terms of Riemann mappings to standard slit domains.     

The Carathéodory topology for multiply connected domains II

We continue our exposition concerning the Carathéodory topology for multiply connected domains which we began in [Comerford M., The Carathéodory topology for multiply connected domains I, Cent. Eur. J. Math., 2013, 11(2), 322–340] by introducing the notion of boundedness for a family of pointed domains of the same connectivity. The limit of a convergent sequence of n-connected domains which is bounded in this sense is again n-connected and will satisfy the same bounds. We prove a result which establishes several equivalent conditions for boundedness. This allows us to extend the notions of convergence and equicontinuity to families of functions defined on varying domains.     

Carathéodory Convergence and Harmonic Measure

We give several new characterizations of Carathéodory convergence of simply connected domains. We then investigate how different definitions of convergence generalize to the multiply-connected case.

     

Solvability of a class of elastic beam equations with strong Carathéodory nonlinearity

We study the existence of a solution to the nonlinear fourth-order elastic beam equation with nonhomogeneous boundary conditions

Eversive maps of bounded convex domains in ℂ n+1

A duality principle, relating the geometry of the Kobayashi metric with the CR geometry of the boundaries of smoothly bounded, strongly convex domains in ℂ ⁿ⁺¹ is established. A characterization of the holomorphic Jacobi vector fields of those domains is also given.     

Continuity and boundary behavior of the Carathéodory metric

The boundary behavior of the higher-order Carathéodory metrics, the singular Carathéodory metric, and the Azukawa metric near anh-extendable boundary point of a bounded smooth pseudoconvex domain in ℂⁿ are studied. Translated fromMathematicheskie Zametski, Vol. 67, No. 2, pp. 230–240, February, 2000.     

Nonnegative Polynomials and Their Carathéodory Number

In 1888 Hilbert showed that every nonnegative homogeneous polynomial with real coefficients of degree $2d$ in $n$ variables is a sum of squares if and only if $d=1$ (quadratic forms), $n=2$ (binary forms) or $(n,d)=(3,2)$ (ternary quartics). In these cases, it is interesting to compute canonical expressions for these decompositions. Starting from Carathéodory’s Theorem, we compute the Carathéodory number of Hilbert cones of quadratic forms and binary forms.     

The Gleason''''s problem on mixed norm spaces in convex domains

Let Ω be a bounded convex domain with C2 boundary in C2 and for given 0 < p, q ≤∞ and normal weight function (r) let Hp,q, be the mixed norm space on Ω. In this paper we prove that the Gleason's problem (Ω, a, Hp,q,) is solvable for any fixed point a ∈ Ω. While solving the Gleason's problem we obtain the boundedness of certain integral operator on Hp,q,.     

Carathéodory’s Theorem and moduli of local connectivity

We give a quantitative proof of the Carathéodory Theorem by means of the concept of a modulus of local connectivity and the extremal distance of the separating curves of an annulus.     

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.     

Carathéodory, Helly and the Others in the Max-Plus World

Carathéodory’s, Helly’s and Radon’s theorems are three basic results in discrete geometry. Their max-plus or tropical analogues have been proved by various authors. We show that more advanced results in discrete geometry also have max-plus analogues, namely, the colorful Carathéodory theorem and the Tverberg theorem. A conjecture connected to the Tverberg theorem—Sierksma’s conjecture—although still open for the usual convexity, is shown to be true in the max-plus setting.     

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.