Solutions of the $\mathrm{SU}(n+1)$ Toda system from meromorphic functions

Abstract

We consider the $\mathrm{SU}(n+1)$ Toda system on a simply connected domain $\Omega$ in $\mathbb{C}$, the $n=1$ case of which coincides with the Liouville equation $\Delta u+8e^u=0$. A classical result by Liouville says that a solution of this equation on $\Omega$ can be represented by some non-degenerate meromorphic function on $\Omega$. We construct a family of solutions parameterized by $\mathrm{PSL}(n+1,\mathbb{C})/\mathrm{PSU}(n+1)$ for the $\mathrm{SU}(n+1)$ Toda system from such a meromorphic function on $\Omega$, which generalizes the result of Liouville. As an application, we find a new class of solvable $\mathrm{SU}(n+1)$ Toda systems with singular sources via cone spherical metrics on compact Riemann surfaces.