Geodesic
Elliptic functions, Green functions and the mean field equations on tori
Chang-Shou Lin1 ; Chin-Lung Wang2
1Department of Mathematics and Taida Institute of Mathematical Sciences (TIMS)<br/>National Taiwan University<br/>Taipei, Taiwan
2Department of Mathematics<br/>National Taiwan University<br/>No. 1, Sec. 4, Roosevelt Road<br/>Taipei, 10617<br/>Taiwan<br/>and <br/>Department of Mathematics<br/>National Central University<br/>Chung-Li<br/>Taiwan
Annals of mathematics, Tome 172 (2010) no. 2, pp. 911-954
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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.

DOI : 10.4007/annals.2010.172.911

Chang-Shou Lin  1   ; Chin-Lung Wang  2

1 Department of Mathematics and Taida Institute of Mathematical Sciences (TIMS)<br/>National Taiwan University<br/>Taipei, Taiwan
2 Department of Mathematics<br/>National Taiwan University<br/>No. 1, Sec. 4, Roosevelt Road<br/>Taipei, 10617<br/>Taiwan<br/>and <br/>Department of Mathematics<br/>National Central University<br/>Chung-Li<br/>Taiwan 
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Chang-Shou Lin; Chin-Lung Wang. Elliptic functions, Green functions and the mean field equations on tori. Annals of mathematics, Tome 172 (2010) no. 2, pp. 911-954. doi: 10.4007/annals.2010.172.911

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