Toshiyuki SUGAWA

*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.

submission: 30 May 2001

revision: 23 October 2001

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