Solutions for Toda system on Riemann surface with boundary
Consider a compact Riemann surface \((M,g)\) with boundary \(\partial M\), and suppose that the metric is normalized so that the area of \(M\) is one. In this paper, sufficient conditions for the existence of solutions to the Toda system subject to Neumann boundary conditions on \(M\) are found. In variational formulation, solutions to the Toda system are critical points \(u = (u_1,u_2)\) of the functional \[ \begin{aligned} \Phi_{(M_1,M_2)}(u_1,u_2) = & \frac{1}{2}\sum_{i,j=1}^2 \int_M a_{ij} (\nabla u_i \nabla u_j + 2M_iu_j)\,dV_g \\ & - \sum_{i=1}^2 M_i \log\int_M \exp\left(\sum_{j=1}^2 a_{ij}u_j\right)\,dV_g, \end{aligned} \] where \(a_{11} = a_{22} = 2\), and \(a_{12} = a_{21} = -1\). Minimizers exist if the parameters \(M_1,M_2 < 2\pi\) by direct methods, and the paper is concerned with the critical case \(M_1=M_2=2\pi\). The result of the paper is that a minimizer exists in the critical case if either the geodesic curvature \(k_g\) of \(\partial M\) satisfies \(\max_{p \in \partial M}k_g(p) < 0\), or, if \(k_g \leq 0\) everywhere on \(\partial M\) and if \(k_g(p) = 0\) at \(p \in \partial M\), then the Gauss curvature \(K\) of \(M\) satisfies \(K(p) < \pi\).
scientific article; zbMATH DE number 6374582