Title: Universal locally univalent functions and universal conformal metrics with constant curvature,
Abstract: We prove Runge-type theorems and universality results for locally univalent
holomorphic and meromorphic functions. Refining a result of M. Heins, we also
show that there is a universal bounded locally univalent function on the unit
disk. These results are used to prove that on any hyperbolic simply connected
plane domain there exist universal conformal metrics with prescribed constant
curvature.