The Green function is defined as the solution of $\Delta u=-\delta+const$. It turns out that $u$ can have three or five critical points, depending on the modulus of the torus. Surprisingly, this is a recent result (C.-S. Lin and C.-L. Wang, 2010).

We give a new, much simpler proof based on holomorphic dynamics. In the process we discovered a simple one-parametric family of holomorphic dynamical systems, whose parameter space consists of just two hyperbolic components and a smooth curve separating them. Of course the dependence on parameter in this family is not complex-analytic, but just real analytic.