Classification of solutions of a Toda system in \(\mathbb{R}^2\) (Q2778018)

The paper deals with the 2-dimensional Toda system for SU(N+1) of the form NEWLINE\[NEWLINE-\frac{1}{2}\triangle u_i=\sum \limits_{j=1}^Na_{ij}e^{u_j}\tag{1}NEWLINE\]NEWLINE in \({\mathbb R}^2\) for \(i=1,2,\cdots ,N\in \mathbb N\), where \(K=(a_{ij})_{N\times N}\) is the Cartan matrix in SU(N+1). It is shown that any \(C^2\)-solution \(u=(u_1,u_2,\cdots ,u_N)\) of (1) satisfying \(\int \limits_{{\mathbb R}^2}e^{u_i}<\infty\), \(i=1,2,\dots ,N\) has the form NEWLINE\[NEWLINEu_i(z)=\sum \limits_{j=1}^Na_{ij}\log \|\Lambda_j(f)\|^2NEWLINE\]NEWLINE for some rational curve in \(\mathbb{C}\mathbb{P}^N\). The analytic and geometric aspects of the Toda system are also presented.