Meromorphic Extensions of Green's Functions on a Riemann Surface

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.