On the hyperbolic distance of n-times punctured spheres




Toshiyuki Sugawa, Matti Vuorinen, Tanran Zhang

PublisherHEBREW UNIV MAGNES PRESS

2020

Journal d'Analyse Mathématique

JOURNAL D ANALYSE MATHEMATIQUE

J ANAL MATH

25

0021-7670

1565-8538

DOIhttps://doi.org/10.1007/s11854-020-0112-9

https://arxiv.org/abs/1707.05773



The shortest closed geodesic in a hyperbolic surface X is called a systole of X. When X is an n-times punctured sphere C∖^A" role="presentation">C∖ˆA where A⊂C^" role="presentation">A⊂Cˆis a finite set of cardinality n >= 4, we define a quantity Q(A) in terms of cross ratios of quadruples in A so that Q(A) is quantitatively comparable with the systole length of X. We next propose a method to construct a distance function dX on a punctured sphere X which is Lipschitz equivalent to the hyperbolic distance hX on X. In particular, when the construction is based on a modified quasihyperbolic metric, dX is Lipschitz equivalent to hX with a Lipschitz constant depending only on Q(A).



Last updated on 2024-26-11 at 14:46