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

Abstract:

Consider a positive integer $n$ and $\gamma _1>-1,\cdots ,\gamma _n>-1$. Let $D=\{z\in \mathbb {C}:|z|<1\}$, and let $(a_{ij})_{n\times n}$ denote the Cartan matrix of $\frak {su}(n+1)$. Utilizing the ordinary differential equation of $(n+1)$th order around a singular source of ${SU}(n+1)$ Toda system, as discovered by Lin-Wei-Ye [Invent. Math. 190 (2012), pp. 169–207], we precisely characterize a solution $(u_1,\cdots , u_n)$ to the ${SU}(n+1)$ 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}&=\pi \gamma _i\delta _0\text { on } D\\ \frac {\sqrt {-1}}{2}\,\int _{D\backslash \{0\}} e^{u_{i} }{d}z\wedge {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 $1\leq i\leq n$, $0$ represents the cone singularity with angle $2\pi (1+\gamma _i)$ for the metric $e^{u_i}|{d}z|^2$ on $D\backslash \{0\}$, which can be locally characterized by $(n-1)$ non-vanishing holomorphic functions at $0$.