Solutions for Toda systems on Riemann surfaces

TL;DR

The paper analyzes the critical Toda system on a compact Riemann surface by a variational approach, focusing on the N=2 case. It performs a detailed blow-up analysis following Jost–Wang, distinguishing two possible loss-of-compactness scenarios and deriving sharp lower bounds for the energy via capacity arguments and Green’s function asymptotics. By constructing delicate test functions that beat these lower bounds under a Gauss-curvature constraint max K < 2π, it rules out blow-up and proves the existence of a minimizer for the critical functional Φ. The results advance understanding of non-Abelian Chern–Simons/Toda systems on curved surfaces, providing a rigorous existence theory in the critical regime.

Abstract

In this paper, we study the solutions of Toda systems on Riemann surface in the critical case, we prove a sufficient condition for the existence of solutions of Toda systems.