Elliptic functions, Green functions and the mean field equations on tori
Resource
Annals of Mathematics, 172(2), 911-954
arXiv:math/0608358
Journal
Annals of Mathematics
Journal Volume
172
Journal Issue
2
Pages
911-954
Date Issued
2010
Author(s)
Abstract
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.
Type
journal article
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