Entire curves generating all shapes of Nevanlinna currents

Abstract

First, we show that every complex torus $T$ contains some entire curve $g:C\to T$ such that the concentric holomorphic discs ${\left\{g{|}_{{\overline{D}}_r}\right\}}_{r>0}$ can generate all the Nevanlinna/Ahlfors currents on $T$ at cohomological level. This confirms an anticipation of Sibony. Developing further our new method, we can construct some twisted entire curve $f:C\to {\mathrm{CP}}^1\times E$ in the product of the rational curve ${\mathrm{CP}}^1$ and an elliptic curve $E$, such that, concerning Siu's decomposition, demanding any cardinality $\left|J\right|\in Z_{⩾0}\cup \left\{\infty \right\}$ and that $T_{\mathrm{diff}}$ is trivial ($\left|J\right|⩾1$) or not ($\left|J\right|⩾0$), we can always find a sequence of concentric holomorphic discs ${\left\{f{|}_{{\overline{D}}_{r_j}}\right\}}_{j⩾1}$ to generate a Nevanlinna/Ahlfors current $T=T_{\mathrm{alg}}+T_{\mathrm{diff}}$ with the singular part $T_{\mathrm{alg}}={\sum }_{j\in J}{\lambda }_j·\left[C_j\right]$ in the desired shape. This fulfills the missing case where $\left|J\right|=0$ in the previous work of Huynh–Xie. By a result of Duval, each $C_j$ must be rational or elliptic. We will show that there is no a priori restriction on the numbers of rational and elliptic components in the support of $T_{\mathrm{alg}}$, thus answering a question of Yau and Zhou. Moreover, we will show that the positive coefficients ${\left\{{\lambda }_j\right\}}_{j\in J}$ can be arbitrary as long as the total mass of $T_{\mathrm{alg}}$ is less than or equal to 1.