Grafting hyperbolic metrics and Eisenstein series

TL;DR

This paper combines hyperbolic metrics and Eisenstein series to derive explicit expansions for degenerating Riemann surfaces, with applications to Weil-Petersson metric asymptotics and symplectic reduction.

Contribution

It introduces a novel explicit expansion method for hyperbolic metrics using Eisenstein series in degenerating families of Riemann surfaces.

Findings

Asymptotic expansion for Weil-Petersson metric

Explicit local form of symplectic reduction

New connections between hyperbolic metrics and Eisenstein series

Abstract

The family hyperbolic metric for the plumbing variety $\{zw=t\}$ and the non holomorphic Eisenstein series $E(\zeta;2)$ are combined to provide an explicit expansion for the hyperbolic metrics for degenerating families of Riemann surfaces. Applications include an asymptotic expansion for the Weil-Petersson metric and a local form of symplectic reduction.