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arXiv · 2211.15141

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

Yiqian Shi·Chunhui Wei·Bin Xu
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Abstract

We consider the ${\rm SU}(n+1)$ Toda system on a simply connected domain $Ω$ in ${\Bbb C}$, the $n=1$ case of which coincides with the Liouville equation $Δu+8e^u=0$. A classical result by Liouville says that a solution of this equation on $Ω$ can be represented by some non-degenerate meromorphic function on $Ω$. 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 $Ω$, 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.

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Yiqian Shi, Chunhui Wei, Bin Xu. 2022-12-01. Solutions of the ${\rm SU}(n+1)$ Toda system from meromorphic functions. https://arxiv.org/abs/2211.15141

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