The Hilbert-uniformization is real-analytic

Abstract

In \cite{Boed}, C.-F. B\"odigheimer constructed a finite cell-complex $\mf{Par}_{g,n,m}$ and a bijective map $\cH: \mf{Dip}_{g,n,m} \to \mf{Par}_{g,n,m}$ (the Hilbert-uniformization) from the moduli space of dipole functions on Riemann surfaces with $n$ directions and $m$ punctures to $\mf{Par}_{g,n,m}$. In \cite{Boed} and \cite{Eb}, it is proven that $\cH$ is a homeomorphism. The first result of this note is that the space $\mf{Dip}_{g,n,m}$ carries a natural structure of a real-analytic manifold. Our second result is that $\cH$ is real-analytic, at least on the preimage of the top-dimensional open cells of $\mf{Par}_{g,n,m}$.