On the hyperbolic distance of n-times punctured spheres

Toshiyuki Sugawa, Matti Vuorinen, Tanran Zhang

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


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 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 length of X. We next propose a method to construct a distance function dX onapunctured 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).

Original languageEnglish
Pages (from-to)663-687
Number of pages25
JournalJournal d'Analyse Mathematique
Issue number2
Publication statusPublished - 2020 Sept


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