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

By Changshou Lin, Chin-Lung Wang. 2006

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We show that the Green functions on flat tori can have either 3 or 5 critical points only. There does not seem to be any direct method to attack this problem. Instead, we have to employ sophisticated non-linear partial differential equations to study it. We also study the distribution of 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. The general picture is also emerged, though some of the necessary technicality is still to de developed. 1. INTRODUCTION AND STATEMENT OF RESULTS The study of geometric or analytic problems on two dimensional tori is the same as the study of problems on R 2 with doubly periodic data. Such situations occur naturally in sciences and mathematics since early days. The mathematical foundation of elliptic functions was subsequently devel- oped in the 19th century. It turns out that these special functions are rather deep objects by themselves. Tori of different shape may result in very dif- ferent behavior of the elliptic functions and their associated objects. Arith- metic on elliptic curves is perhaps the eldest and the most vivid example. In this paper, we show that this is also the case for certain non-linear par- tial differential equations. Indeed, researches on doubly periodic problems in mathematical physics or differential equations often restrict the study to rectangular tori for simplicity. This leaves the impression that the theory for general tori may resemble much the same way as for the rectangular case. However this turns out to be false. We will show that the solvability of themean field equation dependson the shape of the Green function, which in turn depends on the geometry of the tori in an essential way. Recall that the Green function G(z,w) on a flat torus T = C/Zω1 + Zω2 is the unique function on T × T which satisfies −△zG(z,w) = δw(z) − 1 |T|

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fa3b0a7179272c9137e7a6296b5757be04bca583

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Citations in the corpus

Nondegeneracy of entire solutions of a singular Liouvillle equation (2012, background): “We refer the reader to [11, 12, 16] and references therein for recent developments in this subject and related issues.”
Mean field equations, hyperelliptic curves and modular forms: I (2015, methodology): “3) (for type II) directly, though some simplest cases had been treated in [42, 43] for ρ = 4π, 8π and 12π.”