Related papers: Domains of Holomorphy

We study monodromy of holomorphic motions and show the equivalence of triviality of monodromy of holomorphic motions and extensions of holomorphic motions to continuous motions of the Riemann sphere. We also study liftings of holomorphic…

Given any model category, or more generally any category with weak equivalences, its simplicial localization is a simplicial category which can rightfully be called the "homotopy theory" of the model category. There is a model category…

The main purpose of the present paper is to introduce the notion of squeezing functions of bounded domains and study some properties of them. The relation to geometric and analytic structures of bounded domains will be investigated.…

Let $D_j\subset\Bbb C^{k_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluripolar set, $j=1,...,N$. Put$$X:=\bigcup_{j=1}^N A_1\times...\times A_{j-1}\times D_j\times A_{j+1}\times...\times A_N\subset\Bbb…

We prove a Montel theorem for Hilbert space valued functions, and a non-commutative version of this theorem, by composing with unitaries to achieve convergence.

Let X and Y be bounded multiply connected Lipschitz domains in \R^2. We consider the class H_p (X, Y) of homeomorphisms h : X -> Y in the Sobolev space W^{1,p} (X, \R^2). We prove that the weak and strong closures of H_p (X, Y), 2 \le p<…

The classical results about the boundary values of holomorphic or harmonic functions on a domain $D$ state that under additional integrability assumptions these functions have limits along specific sets approaching boundary. The proofs of…

A domain $G\subset \bar\mathbb C$ is the domain of holomorphy of the generating function of a P\'olya frequency sequence of order $r$ if and only if it satisfies the following conditions: (A) $G$ contains the point $z=0$, (B) $G$ is…

A basic result in the theory of holomorphic functions of several complex variables is the following special case of the work of H. Cartan on the sheaf cohomology on Stein domains ([10], or see [14] or [16] for more modern treatments).

We show that a fairly arbitrary Frechet space topology on the space of holomorphic functions on a domain controls the topology of uniform convergence on compact sets. In fact it turns out that the result we present can be proved more simply…

We investigate the boundary behavior of holomorphic functions with respect to a family of curves in a domain of finite type. This work is a generalization of \u{C}irka's classical result on the unit ball and it supplements the result by…

For $D$, $D'$ analytic polyhedra in $C^n$, it is proven that a biholomorphic mapping $f\colon D\to D'$ extends holomorphically to a dense boundary subset under certain condition of general position. This result is also extended to a more…

We characterise piecewise Boolean domains, that is, those domains that arise as Boolean subalgebras of a piecewise Boolean algebra. This leads to equivalent descriptions of the category of piecewise Boolean algebras: either as piecewise…

We show that the classical kernel and domain functions associated to an n-connected domain in the plane are all given by rational combinations of three or fewer holomorphic functions of one complex variable. We characterize those domains…

We show that a weakly holomorphic modular function can be written as a sum of modular units of higher level. We further find a necessary and sufficient condition for a Siegel modular function of degree $g$ to have neither zero nor pole on…

We give a concrete sufficient condition for a simply-connected domain to be the image of the unit disk under a nonexpansive conformal map. This class of domains is also characterized by having sufficiently dense harmonic measure. The…

We consider a continuous function $f$ on a domain in $\mathbf C^n$ satisfying the inequality that $|\bar \partial f|\leq |f|$ off its zero set. The main conclusion is that the zero set of $f$ is a complex variety. We also obtain removable…

Let $(W,\Pi)$ be a Riemann domain over a complex manifold $M$ and $w_0$ be a point in $W$. Let $\mathbb D$ be the unit disk in $\mathbb C$ and $\mathbb T=\bd\mathbb D$. Consider the space ${\mathcal S}_{1,w_0}({\bar{\mathbb D}},W,M)$ of…

If $V$ is an analytic set in a pseudoconvex domain $\Omega$, we show there is always a pseudoconvex domain $G \subseteq \Omega$ that contains $V$ and has the property that every bounded holomorphic function on $V$ extends to a bounded…

Let $D$ be a strictly pseudoconvex domain and $X$ be a singular analytic set of pure dimension $n-1$ in $C^n$ such that $X\cap D\neq \emptyset$ and $X\cap bD$ is transverse. We give sufficient conditions for a function holomorphic on $D\cap…