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15 June 2007 Cusps and the family hyperbolic metric
Scott A. Wolpert
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Duke Math. J. 138(3): 423-443 (15 June 2007). DOI: 10.1215/S0012-7094-07-13833-X

Abstract

The hyperbolic metric for the punctured unit disc in the Euclidean plane is singular at the origin. A renormalization of the metric at the origin is provided by the Euclidean metric. For Riemann surfaces, there is a unique germ for the isometry class of a complete hyperbolic metric at a cusp. The renormalization of the metric for the punctured unit disc provides a renormalization for a hyperbolic metric at a cusp. For a holomorphic family of punctured Riemann surfaces, the family of (co)tangent spaces along a puncture defines a tautological holomorphic line bundle over the base of the family. The Hermitian connection and Chern form for the renormalized metric are determined. Connections to the works of M. Mirzakhani [Mi1], [Mi2] and L. Takhtajan and P. Zograf [TZ2] and to intersection numbers for the moduli space of punctured Riemann surfaces studied by E. Witten [Wi1], [Wi2] are presented

Citation

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Scott A. Wolpert. "Cusps and the family hyperbolic metric." Duke Math. J. 138 (3) 423 - 443, 15 June 2007. https://doi.org/10.1215/S0012-7094-07-13833-X

Information

Published: 15 June 2007
First available in Project Euclid: 18 June 2007

zbMATH: 1144.14029
MathSciNet: MR2322683
Digital Object Identifier: 10.1215/S0012-7094-07-13833-X

Subjects:
Primary: 14H60 , 30F60
Secondary: 14H15 , 32G15

Rights: Copyright © 2007 Duke University Press

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Vol.138 • No. 3 • 15 June 2007
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