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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2302.13068 |
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Table of Contents:
- Consider a positive integer $n$ and $γ_1>-1,\cdots,γ_n>-1$. 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}&=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 $1\leq i\leq n$, $0$ represents the cone singularity with angle $2π(1+γ_i)$ 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 $0$.