Uniqueness and globality of the Liouville formula for entire solutions of ((partial deriv)~2logλ)/((partial deriv)z(partial deriv)z) + λ/2 = 0

摘要

In this paper we study the Liouville equation Δu = e~(-2u) in dimension two. We prove that an entire solution u determines, up to a subgroup of Mobius 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.