Equilibrium point of Green’s function for the annulus and Eisenstein series

Abstract

We study the motion of the equilibrium point of Green’s function and give an explicit parametrization of the unique zero of the Bergman kernel of the annulus. This problem is reduced to solving the equation ℘ ( z , τ ) = − π 2 3 E 2 ( τ ) \wp (z,\tau )= -\frac {\pi ^2}{3}E_2(\tau ) , where E 2 ( τ ) E_2(\tau ) is the usual Eisenstein series.