Solutions for Toda system on Riemann surface with boundary
type: Item
instance of: scholarly article
title: Solutions for Toda system on Riemann surface with boundary
zbMATH Open document ID: 1331.35127
author: Xiaobao Zhu
publication date: 2014-11-27T00:00:00Z
review text: 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 \(K(p) < \pi\).
reviewed by: Thomas Krainer
Mathematics Subject Classification ID: 35J57
Mathematics Subject Classification ID: 35J50
Mathematics Subject Classification ID: 58J32
zbMATH DE Number: 6374582
zbMATH Keywords: Riemann surface with boundary
zbMATH Keywords: Toda system
zbMATH Keywords: Neumann boundary condition
MaRDI profile type: Publication
full work available at URL: s10114-011-9532-x
OpenAlex ID: W2110502689
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DOI: 10.1007/S10114-011-9532-X
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 6374582