Consider a positive integer $n$ and $\gamma_1>-1,\cdots,\gamma_n>-1$. Let $D=\{ z\in { \Bbb C}:|z|<1\}$, 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 ${ \rm SU}(n+1)$ 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 ${ \rm SU}(n+1)$ 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\,\,{ \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 $1\leq i\leq n$, $0$ represents the cone singularity with angle $2\pi(1+\gamma_i)$ for the metric $e^{ u_i}|{ \rm d}z|^2$ on $D\backslash\{ 0\}$, which can be locally characterized by $(n-1)$ non-vanishing holomorphic functions at $0$.
Categories: math.AP math.CV math.DG