On the hyperbolic length and quasiconformal mappings

Let ϕ : R → S be a K-quasiconformal mapping of a hyperbolic Riemann surface R to another S. It is important to see how the hyperbolic structure is changed by ϕ. S. Wolpert (1979, The length spectrum as moduli for compact Riemann surfaces. Ann. of Math. 109, 323-351) shows that the length of a closed...

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Bibliographic Details
Published inComplex variables, theory & application Vol. 50; no. 2; pp. 123 - 130
Main Author Shiga, Hiroshige
Format Journal Article
LanguageEnglish
Japanese
Published Taylor & Francis Group 10.02.2005
Subjects
AMS 2000 Mathematics Subject Classifications: Primary 30F40
Hyperbolic geometry
Quasiconformal mapping
Riemann surfaces
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ISSN0278-1077
1563-5066
DOI10.1080/02781070412331328206

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Summary:Let ϕ : R → S be a K-quasiconformal mapping of a hyperbolic Riemann surface R to another S. It is important to see how the hyperbolic structure is changed by ϕ. S. Wolpert (1979, The length spectrum as moduli for compact Riemann surfaces. Ann. of Math. 109, 323-351) shows that the length of a closed geodesic is quasi-invariant. Recently, A. Basmajian (2000, Quasiconformal mappings and geodesics in the hyperbolic plane, in The Tradition of Ahlfors and Bers, Contemp. Math. 256, 1-4) gives a variational formula of distances between geodesics in the upper half-plane. In this article, we improve and generalize Basmajian's result. We also generalize Wolpert's formula for loxodromic transformations.
ISSN:0278-1077
1563-5066
DOI:10.1080/02781070412331328206