Estimates of hyperbolic metric with applications to Teichm\"uller spaces

Abstract

We show a monotonicity property of the hyperbolic metric of a punctured rectangular torus. We will then deduce a lower estimate of the hyperbolic metric of the domain $\mathbb{C}\setminus\{0,1\}.$ We also determine the value of hyperbolic sup-norm of standard quadratic differentials on the once-punctured square torus or the symmetric four-times punctured sphere. This enables us to calculate numerically the inner and outer radii of the Bers embedding of the corresponding Teichm\"uller space under the hypothesis of some conjectural properties of it.

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