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In \cite{BeEr}, Bergweiler and Eremenko computed the number of criticalpoints of the Green's function on a torus by investigating the dynamics of acertain family of antiholomorphic meromorphic functions on tori. They alsoobserved that hyperbolic maps are dense in this family of meromorphic functionsin a rather trivial way. In this paper, we study the parameter space of thisfamily of meromorphic functions, which can be written as antiholomorphicperturbations of Weierstrass Zeta functions. On the one hand, we give acomplete topological description of the hyperbolic components and theirboundaries, and on the other hand, we show that these sets admit naturalparametrizations by associated dynamical invariants. This settles a conjecture,made in \cite{LW}, on the topology of the regions in the upper half plane$\mathbb{H}$ where the number of critical points of the Green's functionremains constant.

Antiholomorphic perturbations of Weierstrass Zeta functions and Green's function on tori