The problem of product of conformal radii of nonoverlapping domains

Let a₁, a₂, a₃, b be distinct points in and let D be the family of all triples of nonoverlapping domains D₁, D₂, D₃ in \ {b} such that a_k∈ D_k, k=1,2,3. For this family we consider the problem on the maximum of the functional I=R₁R₂R₃, where R_k=R(D_k, a_k) is the conformal radius of D_k with respect to a_k. Geometrical properties of the extremal triple of domains are described. We prove that the maximum of I monotonically depends on the position of the point b and find the maximum in some special cases Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 212, 1994, pp. 114–128 Translated by N. Yu. Netsvetaev     

Maximum of a conformal,invariant in the problem of nonoverlapping domains

The paper is devoted to the well-known circle of problems on the maximum of the products of powers of conformal radii of nonoverlapping domains. Let a₁, ..., a_n be distinct points of and let D₁, ..., D_n be a system of simply connected domains in, pairwise disjoint and such that a_kD_k, k=1, ..., n. By R(D_ka_k) we denote the conformal radius of the domain D_k relative to the point a_k. One considers the problem on the maximum of the product in the family of all indicated systems of domains, under the condition that a₁, ..., a_n runs over all systems of distinct points in (n4) and one finds the geometric characteristic of the extremal configurations of this problem in terms of the associated quadratic differential.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 112, pp. 172–183, 1981.     

Problem of the maximum of the product of the conformal radii of nonoverlapping domains in a circle

Let a_k, k=1,2,3, be distinct points of the circle U={z:¦z¦<1}, a_3+k=1/¯a_k, k=1,2,3. Let D₁,...,D₆ be a system of nonoverlapping simply connected domains D₁,...,D₆ on,a_k D_k, k=1,...,6. Let R(D_k,a_k) be the conformal radius of the domain D_k with respect to the point a_k. One formulates the following theorem. For any points a_k U, k=1,2,3, and any system of the indicated domains one has the sharp inequality One points out all the cases when equality prevails in (1). One indicates the main steps of the proof of this theorem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 99–113, 1983.     

On a problem of nonoverlapping domains

We solve the problem of finding the range E of some functional on the class of pairs of functions univalent in the system of the disk and the interior of the disk for the arbitrary parameters characterizing the functional. We prove that E is connected and bounded. Using the method of internal variations and the parametric method, we find the equation of the boundary of E. The obtained results extend Lebedev’s study [1].     

Nayatani's metric and conformal transformations of a Kleinian manifold

According to Schoen and Yau (1988), an extensive class of conformally flat manifolds is realized as Kleinian manifolds. Nayatani (1997) constructed a metric on a Kleinian manifold which is compatible with the canonical flat conformal structure. He showed that this metric has a large symmetry if is a complete metric. Under certain assumptions including the completeness of , the isometry group of coincides with the conformal transformation group of . In this paper, we show that may have a large symmetry even if is not complete. In particular, every conformal transformation is an isometry when corresponds to a geometrically finite Kleinian group.

     

The maximum of the product of conformal, radii of nonoverlapping domains in the disk

Two problems on the extremal decomposition of the unit disk are considered. It is shown that the associated quadratic differentials in these problems have at least two distinct zeros on the unit circle. Bibliography: 7 titles.Dedicated to the 90th anniversary of G. M. Goluzin's birthTranslated fromZapiski Nauchnykh Seminarov POMI, Vol. 237, 1997, pp. 105–118.     

On conformal transformations of kropina metric

M. Hashiguchi [3] has studied the conformal theory of Finsler spaces. The theory of Kropina metric was investigated by L. Berwald [1] and V. K. Kropina [4]. The purpose of the present paper is to establish the conformal theory of Kropina metric. In this paper the transformation formulae for the difference tensor D _ik ⁱ (x, ) and Cartan's connection coefficients _k ^*i (x, ) have been obtained.     

Maximum of the product of the conformal radii of four nonoverlapping domains

Let a_k, k=1,...,4, be given distinct points of . Let D_k, k=1,...,4, be a system of simply connected domains in the closed plane such that a_kD_k, k=1,...,4, D_kD=for k,=1, ...,4, k. Let R(D_k,a_k) be the conformal radius of the domain D_k relative to the point a_k. In this paper we obtain an explicit expression for the maximum of the product in the family of all indicated system of domains in terms of elliptic functions. In the proof one makes directly use of the property of the extremal family of domains of the problem under consideration in terms of the associated quadratic differential.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 100, pp. 146–165, 1980.     

Approximation of conformal mappings by circle patterns

A circle pattern is a configuration of circles in the plane whose combinatorics is given by a planar graph G such that to each vertex of G corresponds a circle. If two vertices are connected by an edge in G, the corresponding circles intersect with an intersection angle in (0, π). Two sequences of circle patterns are employed to approximate a given conformal map g and its first derivative. For the domain of g we use embedded circle patterns where all circles have the same radius decreasing to 0 and with uniformly bounded intersection angles. The image circle pattern has the same combinatorics and intersection angles and is determined from boundary conditions (radii or angles) according to the values of g′ (|g′| or arg g′). For quasicrystallic circle patterns the convergence result is strengthened to C ^∞-convergence on compact subsets.