Meromorphic Extensions of Green's Functions on a Riemann Surface

TL;DR

This paper introduces Green's functions with extended meromorphicity (GEM forms) on Riemann surfaces, providing explicit constructions, a reformulation of the Bers map, and insights into moduli space variations.

Contribution

It constructs GEM forms that are meromorphic in both variables, reformulates the Bers map, and offers explicit operators for moduli space variations on Riemann surfaces.

Findings

GEM forms are meromorphic in both variables with a simple pole at x=y.

A reformulation of the Bers map is provided.

An explicit differential operator describes moduli space variations.

Abstract

For a Riemann surface of genus $g\ge 2$ there exists a unique Green's function $G_{N}(x,y)$ which transforms as a weight $N\ge 2$ form in $x$ and a weight $1-N$ form in $y$ and is meromorphic in $x$, with a unique simple pole at $x=y$, but is not meromorphic in $y$. For a Schottky uniformized Riemann surface we consider meromorphic extensions of $G_{N}(x,y)$ called Green's Functions with Extended Meromorphicity or GEM forms. GEM forms are meromorphic in both $x$ and $y$ with a unique simple pole at $x=y$, transform as weight $N\ge 2$ forms in $x$ but as weight $1-N$ quasiperiodic forms in $y$. We give a reformulation of the bijective Bers map and describe a choice of GEM form with an associated canonical basis of normalized holomorphic $N$-forms. We describe an explicit differential operator constructed from $N=2$ GEM forms giving the variation with respect to moduli space parameters of a punctured Riemann surface. We also describe a new expression for the inverse Bers map.