EXACT NUMBER AND NON-DEGENERACY OF CRITICAL POINTS OF MULTIPLE GREEN FUNCTIONS ON RECTANGULAR TORI

Let E-tau := (C/(Z + Z tau) be a flat torus and G(z; tau) be the Green function on E-tau. Consider the multiple Green function G(n) on (E-tau)(n): G(n) (z(1), ..., z(n); tau) := Sigma(i<j) G(z(i) - z(j); tau) - n Sigma(n)(i=1) G(z(i); tau). We prove that for tau is an element of iR >(0), i.e. E-tau is a rectangular torus, G(n) has exactly 2n + 1 critical points modulo the permutation group S-n and all critical points are non-degenerate. More precisely, there are exactly n (resp. n + 1) critical points a's with the Hessian satisfying (-1)(n) det D(2)G(n) (a; tau) < 0 (resp. > 0). This confirms a conjecture in [4]. Our proof is based on the connection between G(n) and the classical Lame equation from [4, 19], and one key step is to establish a precise formula of the Hessian of critical points of G(n) in terms of the monodromy data of the Lame equation. As an application, we show that the mean field equation Delta u + e(u) = rho delta(0) on E-tau has exactly n solutions for 8 pi n - rho > 0 small, and exactly n + 1 solutions for rho - 8 pi n > 0 small.