Cusps and the family hyperbolic metric

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