Green's function and anti-holomorphic dynamics on a torus

TL;DR

This paper provides a new proof that the Green function on a torus has either three or five critical points, using anti-holomorphic dynamics, and explores related dynamical systems with hyperbolic components.

Contribution

It introduces a novel, simplified proof of the critical point count for the Green function on a torus and studies a family of anti-holomorphic dynamical systems with specific parameter space properties.

Findings

Green function has 3 or 5 critical points depending on the torus modulus

A family of anti-holomorphic systems with hyperbolic components and separating curves

New proof technique using anti-holomorphic dynamics

Abstract

We give a new, simple proof of the fact recently discovered by C.-S. Lin and C.-L. Wang that the Green function of a torus has either three or five critical points, depending on the modulus of the torus. The proof uses anti-holomorphic dynamics. As a byproduct we find a one-parametric family of anti-holomorphic dynamical systems for which the parameter space consists only of hyperbolic components and analytic curves separating them.