Holomorphic mappings of a strip into itself with bounded distortion at infinity

A class of holomorphic self-mappings of a strip which is symmetric with respect to the real axis is studied. It is required that the mappings should boundedly deviate from the identity transformation on the real axis. Distortion theorems for this class of functions are obtained, and domains of univalence are found that arise for certain values of the parameter characterizing the deviation of the mappings from the identity transformation on the real axis.     

Equivalence of mappings at infinity

We give, in terms of the ?ojasiewicz inequality, a sufficient condition for germs of $C^2$ mappings at infinity to be isotopical.     

Topology at infinity of polynominal mappings and Thom regularity condition

We consider polynomial mappings which have atypical fibres due to the asymptotic behavior at infinity. Fixing some proper extension of the polynomial mapping, we study the localizability at infinity of the variation of topology of fibres and the possibility of interpreting local results at infinity into global results. We prove local and global Bertini–Sard–Lefschetz type statements for noncompact spaces and nonproper mappings and we deduce results on the homotopy type or the connectivity of the fibres of polynomial mappings.     

Rigidity of conformally compact manifolds with the round sphere as the conformal infinity

In this paper, we prove that under a lower bound on the Ricci curvature and an assumption on the asymptotic behavior of the scalar curvature, a complete conformally compact manifold whose conformal boundary is the round sphere has to be the hyperbolic space. It generalizes similar previous results where stronger conditions on the Ricci curvature or restrictions on dimension are imposed.     

On the dynamics near infinity of some polynomial mappings in

We construct the Green current for a random iteration of horizontal-like mappings in . This is applied to the study of a polynomial map with the following properties: i. infinity is f-attracting; ii. f contracts the line at infinity to a point not in the indeterminacy set. We study for such mappings the escape rates near infinity, i.e. the set of possible values of the function We show in particular that the set of possible values can contain an interval. On the other hand the Green current T of f can be decomposed into pieces associated to an itinerary defined by the indeterminacy points. This allows us to prove that exists ||T||-a.e. and we give its value in terms of explicit quantities depending on f.     

Special conformal mappings in the general theory of relativity

Some special conformal mappings of relativistic spaces are studied. The amount of arbitrariness with which one can describe a nontrivial special conformai mapping is determined. Necessary and sufficient conditions are found under which a space-timeV admits a special conformal mapping to space-time ¯V in which the metric tensor of bothV and ¯V satisfy the Einstein equations with the energy-momentum tensor of an ideal fluid having a geodesic velocity field.It is proved that such spaces are spaces of standard cosmological models.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 43–50.     

On mappings related to the gradient of the conformal radius

We establish a criterion for the gradient ?R(D, z) of the conformal radius of a convex domain D to be conformal: the boundary ?D must be a circle. We obtain estimates for the coefficients K(r) for the K(r)-quasiconformal mappings ?R(D, z), D(r) ? D, 0 < r < 1, and supplement the results of Avkhadiev and Wirths concerning the structure of the boundary under diffeomorphic mappings of the domain D.     

A linear operator associated with the Mittag-Leffer function and related conformal mappings

In the present paper, we introduce a linear operator associated with the Mittag-Leffler function. Some convolution properties of meromorphic functions involving this operator are given.