Two characters on one punctured Riemann surface

TL;DR

This paper develops an abstract framework for coupled period-realization of meromorphic 1-forms on punctured Riemann surfaces, enabling the realization of prescribed pairs under certain conditions, and abstracting a classical extremal-length minimization method.

Contribution

It introduces a new abstract framework for realizing pairs of meromorphic differentials on punctured Riemann surfaces, generalizing the Weber–Wolf extremal-length method.

Findings

Existence of points in the character domain corresponding to desired differential pairs.

Framework applies under Teichmüller-regularity, degeneration detection, and pushability conditions.

Abstracts a classical extremal-length minimization technique for minimal surface construction.

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.