Related papers: A set on which the local Lojasiewicz exponent is a…

In this paper, we give a general boundary Schwarz lemma for holomorphic mappings between unit balls in any dimensions. It is proved that if the mapping $f\in C^{1+\alpha}$ at $z_0\in \partial \mathbb B^n$ with $f(z_0)=w_0\in \partial…

Let $\Gamma $ be a $C^\infty $ curve in $\Bbb{C}$ containing 0; it becomes $\Gamma_\theta $ after rotation by angle $\theta $ about 0. Suppose a $C^\infty $ function $f$ can be extended holomorphically to a neighborhood of each element of…

The general theme of this note is illustrated by the following theorem: Theorem 1. Suppose $K$ is a compact set in the complex plane and 0 belongs to the boundary $\partial K$. Let ${\cal A}(K)$ denote the space of all functions $f$ on $K$…

We give an expression for the {\L}ojasiewicz exponent of a wide class of n-tuples of ideals $(I_1,..., I_n)$ in $\O_n$ using the information given by a fixed Newton filtration. In order to obtain this expression we consider a reformulation…

The class of metrizable spaces $M$ with the following approximation property is introduced and investigated: $M\in AP(n,0)$ if for every $\e>0$ and a map $g\colon\I^n\to M$ there exists a 0-dimensional map $g'\colon\I^n\to M$ which is…

Let f be a proper holomorphic mapping between bounded domains D and D' in C^2. Let M, M' be open pieces on the boundaries of D and D' respectively, that are smooth, real analytic and of finite type. Suppose that the cluster set of M under f…

A theorem of A. Ostrowski describing meromorphic functions f such that the family {f(kz):k in C*} is normal, is generalized to holomorphic maps from $C*$ to a projective space.

Let U be the open unit disc in C and let B be the open unit ball in C^2. We prove that every discrete subset of B is contained in the range f(U) of a complete, proper holomorphic embedding f:U-->B. Here the completeness of f means that for…

In this paper, we prove the main properties of the set of fixed points of an holomorphic map from a bounded domain in C^n into itself.

The purpose of this article is to prove a generalisation of the Besicovitch-Federer projection theorem about a characterisation of rectifiable and unrectifiable sets in terms of their projections. For an $m$-unrectifiable set…

We prove a boundary version of the open mapping theorem for holomorphic maps between strongly pseudoconvex domains. That is, we prove that the local image of a holomorphic map $f:D\to D'$ close to a boundary regular contact point $p\in \de…

A classical result in the theory of Loewner's parametric representation states that the semigroup $\mathfrak U_*$ of all conformal self-maps $\phi$ of the unit disk $\mathbb{D}$ normalized by $\phi(0) = 0$ and $\phi'(0) > 0$ can be obtained…

We show that any function $f:\mathbb{H}^n\to\mathbb{H}$ with $f(z+c)=f(z)+c$, $z\in\mathbb{H}^n$, for some $c>0$ has a property that any limit function of a family $\{\frac{f(tz)}{t}\}_{t>0}$ when $t\to\infty$ is linear.

Let $X\subset \mathbb{C}^n$ be a smooth irreducible affine variety of dimension $k$ and let $F: X\to \mathbb{C}^m$ be a polynomial mapping. We prove that if $m\ge k$, then there is a Zariski open dense subset $U$ in the space of linear…

Let $K$ be a closed polydisc or ball in $\C^n$, and let $Y$ be a quasi projective algebraic manifold which is Zariski locally equivalent to $\C^p$, or a complement of an algebraic subvariety of codimension $\ge 2$ in such manifold. If $r$…

We prove a local Nullstellensatz with parameter for a continuous family of c-holomorphic functions with an effective exponent independent of the parameter: the local degree of the cycle of zeroes of the central section section. We assume…

We show that each pseudoconvex domain $\Omega\subset {\mathbb C}^n$ admits a holomorphic map $F$ to ${\mathbb C}^m$ with $|F|\le C_1 e^{C_2 \hat{\delta}^{-6}}$, where $\hat{\delta}$ is the minimum of the boundary distance and…

Let $M$ be a complete complex Hermitian manifold with metric $E_{M}$ and let $\varphi: [0,\infty)\rightarrow (0,\infty)$ be positive function such that $$\gamma_r=\sup\limits_{r\leq a<b}\left|(\varphi(a)-\varphi(b))/(a-b)\right|\leq C,~r\in…

\noindent Let $I$ be an ideal of the ring of formal power series $\bK[[x,y]]$ with coefficients in an algebraically closed field $\bK$ of arbitrary characteristic. Let $\Phi$ denote the set of all parametrizations…

We show that the equality $m_1(f(x))=m_2(g(x))$ for $x$ in a neighborhood of a point $a$ remains valid for all $x$ provided that $f$ and $g$ are open holomorphic maps, $f(a)=g(a)=0$ and $m_1,$ $m_2$ are Minkowski functionals of bounded…