Normal families of holomorphic functions and mappings on a Banach space

The authors lay the foundations for the study of normal families of holomorphic functions and mappings on an infinite-dimensional normed linear space. Characterizations of normal families, in terms of value distribution, spherical derivatives, and other geometric properties are derived. Montel-type theorems are established. A number of different topologies on spaces of holomorphic mappings are considered. Theorems about normal families are formulated and proved in the language of these various topologies. Normal functions are also introduced. Characterizations in terms of automorphisms and also in terms of invariant derivatives are presented.     

Invariant holomorphic extension in several complex variables

Two fundamental problems on the invariant holomorphic extensions have been posed, which are naturally arose from our solution of the extended future tube conjecture and closely and deeply related to the general theory of Stein manifolds due to Cartan-Serre. In this paper, the relationship is presented between the two problems, the motivation of considering the problems, and the methods to approach the problems. We have also posed some questions and conjectures related to this two problems. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

Existence of K-limits of holomorphic maps

Let D be a complete hyperbolic domain in ⁿ, n >1, and N a compact Hermitian manifold. We prove a criterion for the existence of the K-limit of an arbitrary holomorphic mapf: D N at an arbitrary boundary point D under the condition that f has the corresponding radial limit at this point.Translated from Matematicheskie Zametki, vol. 77, no. 4, 2005, pp. 509–514.Original Russian Text Copyright © 2005 by P. V. Dovbush.This revised version was published online in April 2005 with a corrected issue number.     

Approximating fixed points of holomorphic mappings in the Hilbert ball

We approximate fixed points of holomorphic and ρ-nonexpansive self-mappings of the Hilbert ball using both continuous and discrete schemes.     

Schwarz-Pick estimates for holomorphic mappings from the polydisk to the unit ball

In this paper, Schwarz-Pick estimates of arbitrary order partial derivatives for holomorphic mappings from the polydisk to the unit ball are presented. We generalize the early work on Schwarz-Pick estimates of higher order derivatives for bounded holomorphic functions on the unit disk and the polydisk.     

Some boundary properties of holomorphic functions of several complex variables

A local uniqueness theorem and analogs of the theorem on removable singularities under the hypothesis of boundedness are proved for functions satisfying the tangential Cauchy-Riemann conditions on hypersurfaces in Cⁿ. The results can be interpreted as giving certain boundary properties of holomorphic functions of several complex variables.Translated from Matematicheskie Zametki, Vol. 23, No. 6, pp. 839–844, June, 1978.In conclusion I express my thanks to B. V. Shabat for his contant interest in this work.     

Vortex equation in holomorphic line bundle over complex manifold

In this note we study the vortex equation in holomorphic line bundle over non-Kähler complex manifolds. We prove a existence theorem to that equation by means of the upper and lower solution method to some Kazdan-Warner type equation.     

A boundary uniqueness theorem for holomorphic functions of several complex variables

If D ? Cⁿ is a region with a smooth boundary and M ? ?D is a smooth manifold such that for some point p ∈ M the complex linear hull of the tangent plane T_p(M) coincides with Cⁿ, then for each functionf ε A(D) the conditionf¦M=0 implies thatf=0 in D.     

A uniqueness theorem for meromorphic mappings in several complex variables

In this paper, we give a uniqueness theorem for meromorphic mappings from ?n into ?N(?) with rank≥μregardless of multiplicities.     

Multiplicities of fixed points of holomorphic maps in several complex variables

Let Δ^υ be the unit ball in ℂ^υ with center 0 (the origin of υ) and let F:Δ^υ→ℂ^υbe a holomorphic map withF(0) = 0. This paper is to study the fixed point multiplicities at the origin 0 of the iteratesF ⁱ =F∘⋯∘F (i times),i = 1,2,.... This problem is easy when υ = 1, but it is very complicated when υ > 1. We will study this problem generally.     

Bounded holomorphic functions and maps with some boundary behavior

Let Ω be a domain of . As a study of boundary behavior of functions and maps in Ω, we consider “the linear cluster set,” which is the cluster set along segment terminating at a boundary point of Ω. We prove that there exist bounded holomorphic functions and maps defined in Ω which have the linear cluster sets of positive measure at every point of a discrete subset of the boundary of Ω under some conditions.