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 $\hat{I}{”}_C\subset \mathrm{Hom}(\hat{I}{“}_{g,n},\mathbb{C}{)}^2$ with a scale--fixed slice $\hat{I}{”}_C^{\mathrm{sc}}$. Assuming Teichmüller--regularity, degeneration detection, and pushability, we prove that there is point in $\hat{I}{”}_C^{\mathrm{sc}}$ 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.

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