Uniqueness and globality of the Liouville formula for entire solutions of $ {\partial^{2}\log \lambda \over \partial z \partial \overline z} + {\lambda \over 2} = 0 $

Abstract.

In this paper we study the Liouville equation $ \Delta u = e^{-2u} $ in dimension two. We prove that an entire solution u determines, up to a subgroup of Möbius transformations, a unique entire meromorphic function which generates u globally, obtaining in the process the Liouville's formula. Our method is based on global solutions of a linear system of partial differential equations, contrasting to Liouville's proof of his formula, which is clearly local.