Entire curves producing distinct Nevanlinna currents (2309.04690v2)

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.