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Two characters on one punctured Riemann surface

Pradip Kumar

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

Two characters on one punctured Riemann surface

Abstract

We develop an abstract framework for coupled period--realization of meromorphic --forms on punctured Riemann surfaces. A configuration datum gives the combinatorics and determines a restricted character domain with a scale--fixed slice . Assuming Teichmüller--regularity, degeneration detection, and pushability, we prove that there is point in 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.
Paper Structure (21 sections, 16 theorems, 100 equations)

This paper contains 21 sections, 16 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Realization of a single character
  3. Coupled restricted period data and the Weber--Wolf mechanism
  4. Motivating one--parameter model
  5. Terminology and statement of the main theorem
  6. Configuration space of restricted characters
  7. The restricted character domain $\Delta_C$
  8. Admissible curves and extremal--length assignments
  9. Analytic hypotheses on extremal length data
  10. Proof of main theorems
  11. Existence of reflexive points: proof of Theorem \ref{['thm:main']}
  12. Two compatible characters on one surface: proof of Theorem \ref{['thm:twoforms']}
  13. Model cases for the synchronization method
  14. A configuration where the theorem works
  15. From the surface to a configuration space (a scale--fixed slice)
  16. ...and 6 more sections

Key Result

Theorem 1.1

Assume $g\ge 2$. A character $\chi\in\operatorname{Hom}(\Gamma_g,\, \mathbb{C})$ lies in the image of the period map if and only if For $g=1$, a character $\chi\in\operatorname{Hom}(\Gamma_1,\, \mathbb{C})$ arises from a holomorphic abelian differential if and only if $\chi(\Gamma_1)$ is a rank--$2$ lattice and $\mathrm{vol}(\chi)>0$.

Theorems & Definitions (39)

  • Theorem 1.1: Haupt Haupt
  • Theorem 1.2: Chenakkod--Faraco--Gupta CFG
  • Definition 2.1: Configuration datum
  • Definition 2.2: Admissible curve set
  • Theorem 2.4: Existence of a reflexive point
  • Theorem 2.5: Two meromorphic $1$--forms on one punctured surface
  • Definition 3.1: $C$--admissible pairs
  • Definition 3.2: The domain $\Delta_C$
  • Lemma 3.3
  • proof
  • ...and 29 more