A Toda lattice in dimension 2 and Nevanlinna theory

It is shown how to study the 2-D Toda system for SU(n+1) using Nevanlinna theory of meromorphic functions and holomorphic curves. The results generalize recent results of Jost – Wang and Chen – Li. We consider the 2-dimensional open Toda system for SU(n + 1): − 2 ∆uj = −ej−1 + 2ej − ej+1, 1 ≤ j ≤ n, u0 = un+1 = 0, (1) where uj are smooth functions in the complex plane C, and n is a positive integer. This system was recently studied by several authors (see the reference list in [8]). When n = 1 we obtain the Liouville equation −∆u1 = 4e1. (2) Jost and Wang [8] classified all solutions of (1) satisfying the condition ∫ C ej <∞, 1 ≤ j ≤ n. (3) In this paper, such classification will be given for a larger class of solutions, namely those that satisfy