Cusps and the family hyperbolic metric
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 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 work of M. Mirzakhani, L. Takhtajan and P. Zograf, and intersection numbers for the moduli space of punctured Riemann surfaces studied by E. Witten are presented.