Solution to SU(n+1) Toda system generated by spherical metrics

TL;DR

This paper links solutions of the SU(n+1) Toda system on Riemann surfaces to spherical metrics and their monodromy, introducing a new class of solutions with cone singularities.

Contribution

It establishes a correspondence between spherical metrics and Toda system solutions, characterizes solution families via monodromy, and introduces new solutions with cone singularities.

Findings

Solutions generated by spherical metrics include a specific family involving ω.

Characterization of solution families through monodromy groups.

Discovery of new solutions with cone singularities on compact Riemann surfaces.

Abstract

Using the correspondence between solutions to the SU(n+1) Toda system on a Riemann surface and totally unramified unitary curves, we show that a spherical metric $\omega$ generates a family of solutions, including $(i(n+1-i)\omega)_{i=1}^n$. Moreover, we characterize this family in terms of the monodromy group of the spherical metric. As a consequence, we obtain a new solution class to the SU(n+1) Toda system with cone singularities on compact Riemann surfaces, complementing the existence results of Lin-Yang-Zhong (JDG, 114(2):337-391, 2020).