Related papers: Generating all Ahlfors currents by a single entire…

First, inspired by a question of Sibony, we show that in every compact complex manifold $Y$ with certain Oka property, there exists some entire curve $f: \mathbb{C}\rightarrow Y$ generating all Nevanlinna/Ahlfors currents on $Y$, by…

First, we show that every complex torus $\mathbb{T}$ contains some entire curve $g: \mathbb{C}\rightarrow \mathbb{T}$ such that the concentric holomorphic discs $\{g\restriction_{\overline{\mathbb D}_{r}}\}_{r>0}$ can generate all the…

We answer a basic question in Nevanlinna theory that Ahlfors currents associated to the same entire curve may be nonunique. Indeed, we will construct one exotic entire curve $f: \mathbb{C}\rightarrow X$ which produces infinitely many…

We prove that a complex curve charged by a local Ahlfors current is either a disc or an annulus.

We prove that for any complex manifold X, the set of all holomorphic maps from the unit disc to X whose images are everywhere dense in X forms a dense subset in the space of all holomorphic maps from the disc to X. We show by an example…

The Ahlfors-Weill extension of a conformal mapping of the disk is generalized to the lift of a harmonic mapping of the disk to a minimal surface, producing homeomorphic and quasiconformal extensions. The extension is obtained by a…

This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e. those (branched) maps effecting the conformal representation upon the disc of a…

It is shown that every holomorphic map $f$ from a Runge domain $\Omega$ of an affine algebraic variety $S$ into a projective algebraic manifold $X$ is a uniform limit of Nash algebraic maps $f_\nu$ defined over an exhausting sequence of…

We consider a nondegenerate holomorphic map $f: V \mapsto X$ where $(X, \omega)$ is a compact hermitian manifold of dimension higher or equal to $k$ and $V$ is an open connected complex manifold of dimension $k$. In this article we give…

Given a proper map f : M $\rightarrow$ Q, having cell-like point-inverses, from a manifold-without-boundary M onto an ANR Q, it is a much-studied problem to find when f is approximable by homeomorphisms, i.e., when the decomposition of M…

The initial motivation for this paper is to discuss a more concrete approach to an approximation theorem of Axler and Shields, which says that the uniform algebra on the closed unit disc closure(D) generated by z and h --- where h is a…

We prove that every continuous mapping from a separable infinite-dimensional Hilbert space $X$ into $\mathbb{R}^{m}$ can be uniformly approximated by $C^\infty$ smooth mappings {\em with no critical points}. This kind of result can be…

We prove that a sequence of holomorphic discs with totally real boundary conditions has a subsequence that Gromov converges to a stable holomorphic map of genus zero with connected boundary provided that the sequence is bounded and has…

In this paper, we formulate and prove a general compactness theorem for harmonic maps using Deligne-Mumford moduli space and families of curves. The main theorem shows that given a sequence of harmonic maps over a sequence of complex…

We prove the following result: Let $(X,g_0)$ be a complete, connected 4-manifold with uniformly positive isotropic curvature and with bounded geometry. Then there is a finite collection $\mathcal{F}$ of manifolds of the form $\mathbb{S}^3…

The basic result of Oka theory, due to Gromov, states that every continuous map $f$ from a Stein manifold $S$ to an elliptic manifold $X$ can be deformed to a holomorphic map. It is natural to ask whether this can be done for all $f$ at…

We provide a direct construction of Poletsky discs via local arc approximation and a Runge-type theorem by A. Gournay.

We prove that entire conformal curves $\mathbb{R}^n \rightarrow \mathbb{R}^m$ fall into two classes: either the curve is affine or the average energy in a ball is strictly increasing for large radii and diverges to infinity. This rigidity…

We prove the following result: Let $(X,g_0)$ be a complete, connected 4-manifold with uniformly positive isotropic curvature and with bounded geometry. Then there is a finite collection $\mathcal{F}$ of manifolds of the form $\mathbb{S}^3…

Let D be a domain in C^n with smooth boundary, of finite 1-type at a point p in the boundary and such that the closure of D has a basis of Stein Runge neighborhoods. Assume that there exists an analytic disc which intersects the closure of…