Entire Curves Producing Distinct Nevanlinna Currents

Abstract

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 disks $\{f\restriction _{\mathbb {D}(c, r)}\}_{c\in \mathbb {C}, r>0}$. 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 disks $\{g\restriction _{\mathbb {D}_{ r}}\}_{r>0}$ 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. Dedicated to Julien Duval with admiration