Entire curves generating all shapes of Nevanlinna currents

Abstract First, we show that every complex torus contains some entire curve such that the concentric holomorphic discs can generate all the Nevanlinna/Ahlfors currents on at cohomological level. This confirms an anticipation of Sibony. Developing further our new method, we can construct some twisted entire curve in the product of the rational curve and an elliptic curve , such that, concerning Siu's decomposition, demanding any cardinality and that is trivial () or not (), we can always find a sequence of concentric holomorphic discs to generate a Nevanlinna/Ahlfors current with the singular part in the desired shape. This fulfills the missing case where in the previous work of Huynh–Xie. By a result of Duval, each 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 , thus answering a question of Yau and Zhou. Moreover, we will show that the positive coefficients can be arbitrary as long as the total mass of is less than or equal to 1.