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

TL;DR

This paper generalizes Liouville's classical representation of solutions to the Liouville equation to the ${ m SU}(n+1)$ Toda system, constructing solutions from meromorphic functions and applying them to find new solvable systems with singular sources.

Contribution

It introduces a method to construct solutions of the ${ m SU}(n+1)$ Toda system from meromorphic functions, extending Liouville's classical result and applying it to systems with singular sources.

Findings

Constructed a family of solutions parameterized by ${ m PSL}(n+1,{ m C})/{ m PSU}(n+1)$.

Generalized Liouville's representation to higher-rank Toda systems.

Discovered new solvable ${ m SU}(n+1)$ Toda systems with singular sources.

Abstract

We consider the ${\rm SU}(n+1)$ Toda system on a simply connected domain $\Omega$ in ${\Bbb C}$, 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 ${\rm PSL}(n+1,\,{\Bbb C})/{\rm PSU}(n+1)$ for the ${\rm SU}(n+1)$ 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 ${\rm SU}(n+1)$ Toda systems with singular sources via cone spherical metrics on compact Riemann surfaces.