Two characters on one punctured Riemann surface

Abstract

We develop an abstract framework for coupled period--realization of meromorphic $1$--forms on punctured Riemann surfaces. A configuration datum $C$ gives the combinatorics and determines a restricted character domain $\Delta_C\subset\mathrm{Hom}(\Gamma_{g,n},{\mathbb C})^2$ with a scale--fixed slice $\Delta_C^{\mathrm{sc}}$. Assuming Teichm\"uller--regularity, degeneration detection, and pushability, we prove that there is point in $\Delta_C^{\mathrm{sc}}$ which corresponds to a surface carrying two meromorphic differentials realizing any prescribed restricted pair. This abstracts the Weber--Wolf extremal--length minimization method while constructing minimal surfaces.