General tropical convergence of harmonic amoebas

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.