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

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 $ ho_1 in (4kpi , 4(k+1)pi)$, $k in mathbb{N}$, $ ho_2 in (4pi, 8pi )$ 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.