Universal entire curves in projective spaces with slow growth (Zhangchi Chen - Chinese Academy of Science, Beijing)

In 1929, Birkhoff constructed some universal entire functions. An entire function h is universal if any entire function g can be approximated by translations of h, in the topology of uniform convergence on compact sets.

The Nevanlinna characteristic function measures complexity of a transcendental entire function. Dinh-Sibony in their problem list (Problem 9.1) asked the minimal growth of the Nevanlinna characteristic function of universal entire maps in C^1 and P^1. At first glance, universal entire functions looks highly transcendental and highly complicated. However, in the recent work with Song-yan Xie and Dinh Tuan Huynh, we proved that the existence of universal entire curves in any n-dim projective spaces with growth slower than any transcendental growth rate. Our idea is motivated by Runge’s approximation theorem and the theory of Oka manifolds. I will also present some open problems.