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

Abstract

We consider the Toda system of Liouville equations on a compact surface $\Sigma$

$$\left\{\begin{array}{c}-\Delta u_1=2{\rho }_1\left(\frac{h_1{e}^{u_1}}{{\int }_{\Sigma }{h}_1{e}^{u_1}d{V}_g}-1\right)-{\rho }_2\left(\frac{h_2{e}^{u_2}}{{\int }_{\Sigma }{h}_2{e}^{u_2}d{V}_g}-1\right),\hfill \\ & \\ -\Delta u_2=2{\rho }_2\left(\frac{h_2{e}^{u_2}}{{\int }_{\Sigma }{h}_2{e}^{u_2}d{V}_g}-1\right)-{\rho }_1\left(\frac{h_1{e}^{u_1}}{{\int }_{\Sigma }{h}_1{e}^{u_1}d{V}_g}-1\right),\hfill \end{array}\right.$$

which arises as a model for nonabelian Chern–Simons vortices. Here $h_1$ and $h_2$ are smooth positive functions and ${\rho }_1$ and ${\rho }_2$ two positive parameters.

For the first time, the ranges ${\rho }_1\in \left(4k\pi ,4\left(k+1\right)\pi \right)$, $k\in \mathbb{N}$, and ${\rho }_2\in \left(4\pi ,8\pi \right)$ are studied with a variational approach on surfaces with arbitrary genus. We provide a general existence result by using a new improved Moser–Trudinger-type inequality and introducing a topological join construction in order to describe the interaction of the two components $u_1$ and $u_2$.