Elliptic functions, Green functions and the mean field equations on tori

• 成果类型: Article
• 署名作者: Lin, Chang-Shou; Wang, Chin-Lung
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• 发表日期: 2010
• 页码: 911-954
• 关键词: riemann surfaces

摘要

We show that the Green functions on flat tori can have either three or five critical points only. There does not seem to be any direct method to attack this problem. Instead, we have to employ sophisticated nonlinear partial differential equations to study it. We also study the distribution of the number of critical points over the moduli space of flat tori through deformations. The functional equations of special theta values provide important inequalities which lead to a solution for all rhombus tori.