Analytic aspects of the toda system: I. A Moser-Trudinger inequality

In this paper, we analyze solutions of the open Toda system and establish an optimal Moser-Trudinger type inequality for this system. Let Sigma be a closed surface with area 1 and K = (a(ij))(NxN) the Cartan matrix for SU(N + 1), i.e., [GRAPHICS] We show that Phi (M)(u) = [GRAPHICS] has a lower bound in (H-1(Sigma))(N) if and only if M-j less than or equal to 4 pi for j = 1, 2, ..., N. This inequality is optimal. As a direct consequence, if M-j < 4<pi> for j = 1, 2,...,N, Phi (M) has a minimizer u that satisfies [GRAPHICS] (C) 2001 John Wiley & Sons, Inc.