Entire curves producing distinct Nevanlinna currents

TL;DR

The paper demonstrates the existence of entire curves in certain complex manifolds that generate all Nevanlinna currents and constructs examples in elliptic products showing the ability to produce infinitely many distinct currents, revealing high holomorphic flexibility.

Contribution

It proves the existence of entire curves generating all Nevanlinna currents in manifolds with Oka property and constructs explicit examples in elliptic products with infinitely many distinct currents.

Findings

Existence of entire curves generating all Nevanlinna currents in certain manifolds.

Construction of entire curves in elliptic products producing infinitely many distinct currents.

Revealing the high holomorphic flexibility of entire curves in large scale geometry.

Abstract

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 holomorphic discs $\{f\restriction_{\mathbb{D}(c, r)}\}_{c\in \mathbb{C}, r>0}$. Next, we answer positively a question of Yau, by constructing some entire curve $g: \mathbb{C}\rightarrow X$ in the product $X:=E_1\times E_2$ of two elliptic curves $E_1$ and $E_2$, such that by using concentric holomorphic discs $\{g\restriction_{\mathbb{D}_{ r}}\}_{r>0}$ we can obtain infinitely many distinct Nevanlinna/Ahlfors currents proportional to the extremal currents of integration along curves $[\{e_1\}\times E_2]$, $[E_1\times \{e_2\}]$ for all $e_1\in E_1, e_2\in E_2$ simultaneously. This phenomenon is new, and it shows tremendous holomorphic flexibility of entire curves in large scale geometry.