Entire curves producing distinct Nevanlinna currents

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 $f:\mathbb{C}\to Y$ generating all Nevanlinna/Ahlfors currents on $Y$, by holomorphic discs $\{f{\upharpoonright }_{\mathbb{D}(c,r)}{\}}_{c\in \mathbb{C},r>0}$. Next, we answer positively a question of Yau, by constructing some entire curve $g:\mathbb{C}\to X$ 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 $\{g{\upharpoonright }_{{\mathbb{D}}_r}{\}}_{r>0}$ 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.

arXiv commentfinal version

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Cited by in corpus (1)

• Oka-1 manifolds