A topological join construction and the Toda system on compact surfaces of arbitrary genus

TL;DR

This paper studies a Toda system of Liouville equations on compact surfaces of any genus, introducing a topological join construction and improved inequalities to establish existence results for specific parameter ranges.

Contribution

It introduces a novel topological join construction and an improved Moser-Trudinger inequality to analyze the Toda system on surfaces of arbitrary genus.

Findings

Established existence of solutions for parameter ranges on arbitrary genus surfaces.

Developed a new topological join construction to describe component interactions.

Proved a general existence result using variational methods and improved inequalities.

Abstract

We consider a Toda system of Liouville equations defined on a compact surface which arises as a model for non-abelian Chern-Simons vortices. For the first time the range of parameters $\rho_1 \in (4k\pi , 4(k+1)\pi)$, $k \in \mathbb{N}$, $\rho_2 \in (4\pi, 8\pi )$ is studied with a variational approach on surfaces with arbitrary genus. We provide a general existence result by means of a new improved Moser-Trudinger type inequality and introducing a topological join construction in order to describe the interaction of the two components.