Classifying solutions of 𝑆𝑈(𝑛+1) Toda system around a singular source

Abstract

Consider a positive integer n n and γ 1 > − 1 , ⋯ , γ n > − 1 \gamma _1>-1,\cdots ,\gamma _n>-1 . Let D = { z ∈ C : | z | > 1 } D=\{z\in \mathbb {C}:|z|>1\} , and let ( a i j ) n × n (a_{ij})_{n\times n} denote the Cartan matrix of s u ( n + 1 ) \frak {su}(n+1) . Utilizing the ordinary differential equation of ( n + 1 ) (n+1) th order around a singular source of S U ( n + 1 ) {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 , ⋯ , u n ) (u_1,\cdots , u_n) to the S U ( n + 1 ) {SU}(n+1) Toda system { ∂ 2 u i ∂ z ∂ z ¯ + ∑ j = 1 n a i j e u j a m p ; = π γ i δ 0 on D − 1 2 ∫ D ∖ { 0 } e u i d z ∧ d z ¯ a m p ; > ∞ for all i = 1 , ⋯ , n \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 ) (n+1) holomorphic functions that satisfy the normalized condition. Additionally, we demonstrate that for each 1 ≤ i ≤ n 1\leq i\leq n , 0 0 represents the cone singularity with angle 2 π ( 1 + γ i ) 2\pi (1+\gamma _i) for the metric e u i | d z | 2 e^{u_i}|{d}z|^2 on D ∖ { 0 } D\backslash \{0\} , which can be locally characterized by ( n − 1 ) (n-1) non-vanishing holomorphic functions at 0 0 .