Related papers: Proper holomorphic discs in C^2

The Hayman-Wu theorem states that the preimage of a line or circle L under a conformal mapping from the unit disc to a simply-connected domain U has total Euclidean length bounded by an absolute constant. The best possible constant is known…

Let f be holomorphically continuable over the complex plane except for finitely many branch points contained in the unit disk. We prove that best rational approximants to f of degree n, in the L^2-sense on the unit circle, have poles that…

We design an Algorithm to fabricate universal holomorphic maps between any two complex Euclidean spaces, within preassigned transcendental growth rate. As by-products, universal holomorphic maps from $\mathbb{C}^n$ to $\mathbb{CP}^m$…

The main purpose of this paper is to study the concept of normal function in the context of harmonic mappings from the unit disk $\mathbb{D}$ to the complex plane. In particular, we obtain necessary conditions for that a function $f$ to be…

Given a closed complex hypersurface $Z\subset \mathbb{C}^{N+1}$ $(N\in\mathbb{N})$ and a compact subset $K\subset Z$, we prove the existence of a pseudoconvex Runge domain $D$ in $Z$ such that $K\subset D$ and there is a complete proper…

Given a domain of holomorphy $D$ in $\mathbb{C}^N$, $N\geq 2$, we show that the set of holomorphic functions in $D$ whose cluster sets along any finite length paths to the boundary of $D$ is maximal, is residual, densely lineable and…

Let $(\varphi_t)$, $(\phi_t)$ be two one-parameter semigroups of holomorphic self-maps of the unit disc $\mathbb D\subset \mathbb C$. Let $f:\mathbb D \to \mathbb D$ be a homeomorphism. We prove that, if $f \circ \phi_t=\varphi_t \circ f$…

We consider a generic family of polynomial maps $f:=(f_1,f_2):\mathbb{C}^2\rightarrow\mathbb{C}^2$ with given supports of polynomials, and degree $ d(f):=\max (deg f_1, deg f_2)$. We show that the (non-) properness of maps $f$ in this…

We construct automorphisms of $\mathbb{C}^2$, and more precisely transcendental H\'enon maps, with an invariant escaping Fatou component which has exactly two distinct limit functions, both of (generic) rank 1. We also prove a general…

In this paper we study the minimal and maximal $L^{2}$-cohomology of oriented, possibly not complete, Riemannian manifolds. Our focus will be on both the reduced and the unreduced $L^{2}$-cohomology groups. In particular we will prove that…

We show that the only proper-holomorphic self-maps of bounded domains in C^k whose dynamics escape to a strictly pseudoconvex point of the boundary are automorphisms of the euclidean ball. This is a Wong-Rosay type result for a sequence of…

The dynamical behaviour of a transcendental entire function in any periodic component of the Fatou set is well understood. Here we study the dynamical behaviour of a transcendental entire function $f$ in any multiply connected wandering…

We prove that all harmonic maps from $\mathbb R^2$ to $\mathbb S^2$ with finite energy are nondegenerate. That is, for any harmonic map $u$ from $\mathbb R^2$ to $\mathbb S^2$ of degree $m$ (in $\mathbb Z$), all bounded kernel maps of the…

We study pseudoholomorphic discs with boundaries attached to a real hypersurface in an almost complex manifold of dimension 2. We prove that if the hypersurface contains no discs, then they fill a one sided neighborhood of the hypersurface.

In this paper geometric properties of infinitely renormalizable real H\'enon-like maps $F$ in $\R^2$ are studied. It is shown that the appropriately defined renormalizations $R^n F$ converge exponentially to the one-dimensional…

We prove that every circled domain in the Riemann sphere admits a proper holomorphic embedding to C^2. Our methods also apply to circled domains with punctures, provided that all but finitely many of the punctures belong to the closure of…

In this article, we introduce a new family of sense preserving harmonic mappings f in the open unit disk and prove that functions in this family are close-to-convex. We give some basic properties such as coefficient bounds, growth…

Answering an old question, we find a domain X in the complex projective plane CP^2 which admits a strongly plurisubharmonic function, but such that every holomorphic function on X is constant. The domain X can be chosen diffeomorphic to an…

We study the class $HQ(\mathbb{D})$, the set of harmonic quasiconformal automorphisms of the unit disk $\mathbb{D}$ in the complex plane, endowed with the topology of uniform convergence. Several important topological properties of this…

We study the existence of topologically closed complex curves normalized by bordered Riemann surfaces in complex spaces. Our main result is that such curves abound in any noncompact complex space admitting an exhaustion function whose Levi…