Related papers: Factorization of proper holomorphic mappings throu…

In this paper, we provide a class of domains in $\mathbb{C}^3$, such that every holomorphic self-map of that domain either has a fixed point or the sequence of iterates is compactly divergent. In particular, it follows that the symmetrized…

We study the $\bar{\partial}_b$-Neumann problem for domains $\Omega$ contained in a strictly pseudoconvex manifold M^{2n+1} whose boundaries are noncharacteristic and have defining functions depending solely on the real and imaginary parts…

Given a pseudoconvex domain D in C^N, N>1, we prove that there is a holomorphic function f on D such that the lengths of paths p: [0,1]--> D along which Re f is bounded above, with p(0) fixed, grow arbitrarily fast as p(1)--> bD. A…

Let $X$ be a connected, compact complex manifold and $S\subset X$ a separating real hypersurface, so that $X$ decomposes as a union of compact complex manifolds with boundary $\bar X^\pm$. Let $\mathcal{M}$ be the moduli space of $S$-framed…

We give a necessary and sufficient condition for the smooth extension of a diffeomorphism between smooth strictly pseudoconvex domains in four real dimensional almost complex manifolds. The proof is mainly based on a reflection principle…

In this paper we investigate the global behavior of proper holomorphic maps f from the unit disc U={|z|<1} to C^2. The fact that U is transcendental imposes certain restrictions on the image f(U). For instance, f(U) cannot be contained in…

A matrix formalism is proposed for computations based on Picard--Lefschetz theory in a 2D case. The formalism is essentially equivalent to the computation of the intersection indices necessary for the Picard--Lefschetz formula and enables…

An $n$-dimensional Hartogs domain $D_F$ with strongly pseudoconvex boundary can be equipped with a natural Kaehler metric $g_F$. This paper contains two results. In the first one we prove that if $g_F$ is an extremal Kaehler metric then…

We study bounded pseudoconvex domains in complex Euclidean spaces. We find analytical necessary conditions and geometric sufficient conditions for a domain being of trivial Diederich--Forn\ae ss index (i.e. the index equals to 1). We also…

We present some results dealing with the local geometry of almost complex manifolds. We establish mainly the complete hyperbolicity of strictly pseudoconvex domains, the extension of plurisubharmonic functions through generic submanifolds…

We show that if a bounded domain in complex Euclidean space with $\mathcal{C}^{1,1}$ boundary covers a compact manifold, then the domain is biholomorphic to the unit ball.

H-holomorphic maps are a parameter version of J-holomorphic maps into contact manifolds. They have arisen in efforts to prove the existence of higher--genus holomorphic open book decompositions and efforts to prove the existence of finite…

Let $D\subset \C^n,$ $G\subset \C^m$ be pseudoconvex domains, let $A$ (resp. $B$) be an open subset of the boundary $\partial D$ (resp. $\partial G$) and let $X$ be the 2-fold cross $((D\cup A)\times B)\cup (A\times(B\cup G)).$ Suppose in…

In this paper we study the rigidity of proper holomorphic maps $f\colon \Omega\to\Omega'$ between irreducible bounded symmetric domains $\Omega$ and $\Omega'$ with small rank differences: $2\leq \text{rank}(\Omega')<…

Let $G$ be a bounded open subset in the complex plane and let $H^{2}(G)$ denote the Hardy space on $G$. We call a bounded simply connected domain $W$ perfectly connected if the boundary value function of the inverse of the Riemann map from…

We study regularity properties of CR maps in positive codimension valued in pseudoconvex manifolds which carry a nontrivial Levi foliation. We introduce an invariant which can be used to deduce that any sufficiently regular CR map from a…

Suppose that $f: D\to D'$ is a quasiconformal mapping, where $D$ and $D'$ are domains in ${\mathbb R}^n$, and that $D$ is a broad domain. Then for every arcwise connected subset $A$ in $D$, the weak quasisymmetry of the restriction $f|_A:…

In this paper, we prove that the identity map for the smoothly bounded pseudoconvex domain of finite type in $\mathbb{C}^2$ extends to a bi-H\"{o}lder map between the Euclidean boundary and Gromov boundary. As an application, we show the…

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…

Given $n$ distinct points $\mathbf{x}_1, \ldots, \mathbf{x}_n$ in $\mathbb{R}^d$, let $K$ denote their convex hull, which we assume to be $d$-dimensional, and $B = \partial K $ its $(d-1)$-dimensional boundary. We construct an explicit…