On Solutions for Singular Toda System on Riemann Surfaces with Boundary

TL;DR

This paper investigates the existence and multiplicity of solutions to a singular $SU(3)$ Toda system on Riemann surfaces with boundary, using topological and variational methods.

Contribution

It establishes existence results for non-critical parameters and provides conditions for multiple solutions using Morse theory and transversality.

Findings

Existence of solutions when parameters are non-critical and Euler characteristic is less than one.

A sufficient condition for multiple solutions for generic potentials.

Application of Morse inequalities and transversality theorem to Toda systems.

Abstract

This paper studies solutions to a singular $SU(3)$ Toda system with linear source terms on a compact Riemann surface $\Sigma$ with smooth boundaries $\partial\Sigma$. We establish the existence of solutions when the parameters are not critical, assuming that Euler characteristic $\chi(\Sigma)<1$ via analyzing the sublevels. Furthermore, we find a sufficient condition that ensures multiple solutions for generic potentials by Morse inequalities and a transversality theorem.