Classifying solutions of $\mathrm{SU}(n+1)$ Toda system around a singular source

Abstract

Consider a positive integer $n$ and ${\hat{I}}_1^3>-1,\cdots ,{\hat{I}}_n^3>-1$. Let $D=\{z\in \mathbb{C}:|z|<1\}$, and let $(a_{ij}{)}_{n\times n}$ denote the Cartan matrix of $\mathfrak{su}(n+1)$. Utilizing the ordinary differential equation of $(n+1)$th order around a singular source of $\mathrm{SU}(n+1)$ Toda system, as discovered by Lin-Wei-Ye ({\it Invent Math}, {\bf 190}(1):169-207, 2012), we precisely characterize a solution $(u_1,\cdots ,u_n)$ to the $\mathrm{SU}(n+1)$ Toda system \begin{equation*} \begin{cases} \frac{\partial^2 u_i}{\partial z\partial \bar z}+\sum_{j=1}^n a_{ij} e^{u_j}&=πγ_iδ_0\,\,{\rm on}\,\, D\\ \frac{\sqrt{-1}}{2}\,\int_{D\backslash \{0\}} e^{u_{i} }{\rm d}z\wedge {\rm d}\bar z &< \infty \end{cases} \quad \text{for all}\quad i=1,\cdots, n \end{equation*} using $(n+1)$ holomorphic functions that satisfy the normalized condition. Additionally, we demonstrate that for each $1\le i\le n$, $0$ represents the cone singularity with angle $2\ddot{I}€(1+{\hat{I}}_i^3)$ for the metric $e^{u_i}|\mathrm{d}z{|}^2$ on $D\backslash \{0\}$, which can be locally characterized by $(n-1)$ non-vanishing holomorphic functions at $0$.

In this new version, we have added some references, indicated how our results align with those of Bryant, and addressed additional queries raised by the reviewers