TY - JOUR

T1 - On the hyperbolic distance of n-times punctured spheres

AU - Sugawa, Toshiyuki

AU - Vuorinen, Matti

AU - Zhang, Tanran

N1 - Funding Information:
The authors were supported in part by JSPS Grant-in-Aid for Scientific Research (B) 22340025, and NSF of the Higher Education Institutions of Jiangsu Province, China, 17KJB110015.

PY - 2020

Y1 - 2020

N2 - 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).

AB - 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).

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U2 - 10.1007/s11854-020-0112-9

DO - 10.1007/s11854-020-0112-9

M3 - Article

AN - SCOPUS:85089147924

JO - Journal d'Analyse Mathematique

JF - Journal d'Analyse Mathematique

SN - 0021-7670

ER -