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General tropical convergence of harmonic amoebas

Takashi Ichikawa

TL;DR

The paper develops a framework to describe how harmonic amoebas of pointed Riemann surfaces converge tropically to general (not necessarily simple) tropical curves by employing Schottky uniformization for degenerating curves. It constructs exact and imaginary normalized differentials on degenerating families and proves precise asymptotics that link complex-analytic data to tropical-edge weights and phase information. The main contributions include extending Lang's simple-case tropical convergence to nonsimple graphs, providing explicit formulas for imaginary normalized and Abelian differentials in the Schottky setting, and establishing convergence results for both the tropical shape and the phase of harmonic amoebas. These results have potential applications to compactifying the moduli space of pointed Riemann surfaces with tropical curves and to understanding crystallization phenomena in general dimer models.

Abstract

By using Schottky uniformization theory of degenerating algebraic curves, we describe the tropical convergence of harmonic amoebas of pointed Riemann surfaces to tropical curves which are not necessarily simple. We extend Lang's results on the simple tropical convergence based on the Frenchel-Nielsen coordinates to the nonsimple case. Our results are hoped to give contributions in compactifying the moduli space of pointed Riemann surfaces with tropical curves, and in studying crystallization of general dimer models.

General tropical convergence of harmonic amoebas

TL;DR

The paper develops a framework to describe how harmonic amoebas of pointed Riemann surfaces converge tropically to general (not necessarily simple) tropical curves by employing Schottky uniformization for degenerating curves. It constructs exact and imaginary normalized differentials on degenerating families and proves precise asymptotics that link complex-analytic data to tropical-edge weights and phase information. The main contributions include extending Lang's simple-case tropical convergence to nonsimple graphs, providing explicit formulas for imaginary normalized and Abelian differentials in the Schottky setting, and establishing convergence results for both the tropical shape and the phase of harmonic amoebas. These results have potential applications to compactifying the moduli space of pointed Riemann surfaces with tropical curves and to understanding crystallization phenomena in general dimer models.

Abstract

By using Schottky uniformization theory of degenerating algebraic curves, we describe the tropical convergence of harmonic amoebas of pointed Riemann surfaces to tropical curves which are not necessarily simple. We extend Lang's results on the simple tropical convergence based on the Frenchel-Nielsen coordinates to the nonsimple case. Our results are hoped to give contributions in compactifying the moduli space of pointed Riemann surfaces with tropical curves, and in studying crystallization of general dimer models.
Paper Structure (10 sections, 9 theorems, 33 equations)

This paper contains 10 sections, 9 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. $1$-forms on tropical curves
  3. Schottky uniformization and Abelian differentials
  4. Schottky uniformized deformations
  5. Abelian differentials
  6. General tropical limits of harmonic amoebas
  7. Imaginary normalized differentials
  8. Tropical convergence of integrals
  9. Proof of Theorem 1.1
  10. Proof of Theorem 1.2

Key Result

Theorem 1.1

(cf. L for simple tropical curves) Let $\{ \mathcal{R}_{s} \}_{s>0}$ be a family of $n$-pointed Riemann surfaces of genus $g$ as deformations of $C_{0}$ whose deformation parameters $y_{e}$$(e \in E_{G})$ defined in (3.1) below satisfy $|y_{e}| = s^{\ell_{s}(e)}$, where $\lim_{s \downarrow 0} \ell_{ converges to the map $\pi_{R} : C \rightarrow {\mathbb R}^{m}$ with respect to the Hausdorff distan

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 4.1
  • proof
  • Corollary 4.2
  • Theorem 4.3
  • proof
  • Corollary 4.4
  • proof
  • Proposition 4.5
  • ...and 5 more