Report on the Teichmuller Metric

Let Tgbe the Teichmuller space of compact Riemann surfaces of genus g. Then Tgis the space of conformal structures on a fixed surface W modulo equivalence under conformal maps homotopic to the identity. The Teichmuller modular group Γ is the group of all orientation preserving homeomorphisms of W on...

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Published inProceedings of the National Academy of Sciences - PNAS Vol. 65; no. 3; pp. 497 - 499
Main Author Royden, H. L.
Format Journal Article
LanguageEnglish
Published United States National Academy of Sciences of the United States of America 01.03.1970
National Acad Sciences
Subjects
Cotangent function
Mathematical theorems
Physical Sciences: Mathematics
Riemann surfaces
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Summary:Let Tgbe the Teichmuller space of compact Riemann surfaces of genus g. Then Tgis the space of conformal structures on a fixed surface W modulo equivalence under conformal maps homotopic to the identity. The Teichmuller modular group Γ is the group of all orientation preserving homeomorphisms of W onto itself modulo those which are homotopic to the identity. Each element of Γ induces a biholomorphic map of Tgonto itself, and the present note outlines a proof of the converse statement: Every biholomorphic map of Tgonto itself is induced by an element of Γ . It is first shown that every isometry of Tgwith the Teichmuller metric arises from an element of Γ . The Teichmuller metric is then shown to be the Kobayashi metric for Tgand hence invariant under biholomorphic maps.
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This work was supported in part by the U.S. Army Research Office (Durham) and in part by the National Science Foundation.
ISSN:0027-8424
1091-6490
DOI:10.1073/pnas.65.3.497