On the hyperbolic distance of n-times punctured spheres

2017/07/18 by Sugawa, Toshiyuki, Vuorinen, Matti, Zhang, Tanran
#30C35 (Primary) #30C55 (Secondary) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1707.05773

Abstract

The length of the shortest closed geodesic in a hyperbolic surface X is called the systole of X. When X is an n-times punctured sphere ℂ ∖ A where A ⊂ ℂ is 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 of X. We next propose a method to construct a distance function d_X on a punctured sphere X which is Lipschitz equivalent to the hyperbolic distance h_X on X. In particular, when the construction is based on a modified quasihyperbolic metric, d_X is Lipschitz equivalent to h_X with Lipschitz constant depending only on Q(A).

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