On the hyperbolic distance of n-times punctured spheres
: Toshiyuki Sugawa, Matti Vuorinen, Tanran Zhang
Publisher: HEBREW UNIV MAGNES PRESS
: 2020
: Journal d'Analyse Mathématique
: JOURNAL D ANALYSE MATHEMATIQUE
: J ANAL MATH
: 25
: 0021-7670
: 1565-8538
DOI: https://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).