Green functions, Hitchin's formula, and curvature equations on tori II: Rectangular torus

Abstract Let be the Green function on the flat torus with the singularity at 0. Lin and Wang (Ann. Math. 2010) proved that has either three or five critical points (depending on the choice of ). Here, we study the sum of two Green functions that can be reduced to . In Part I by Chen et al., we proved that for any satisfying in , the number of critical points of belongs to (depending on the choice of ) and each number really occurs. In the Part II of this series, we study the important case with , that is, is a rectangular torus. By developing a completely different approach from Part I, we show the existence of eight real values such that if then has no nontrivial critical points; if then has a unique pair of nontrivial critical points that are always nondegenerate saddle points. This allows us to study the possible distribution of the numbers of critical points of for generic . Applications to the Painlevé VI equation and the curvature equation are also given.