Bers embedding of the Teichmüller space of a once-punctured torus

Abstract

In this note, we present a method of computing monodromies of projective structures on a once-punctured torus or a four-times punctured sphere. This leads to an algorithm numerically visualizing the shape of the Bers embedding of a one-dimensional Teichmüller space. As a by-product, the value of the accessary parameter of a four-times punctured sphere will be calculated in a numerical way as well as generators of a Fuchsian group uniformizing it. Finally, we observe the relation between the Schwarzian differential equation and Heun's differential equation in this special case.

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