Harmonic maps between ideal 2-dimensional simplicial complexes

TL;DR

This paper establishes existence and regularity of energy-minimizing harmonic maps between ideal hyperbolic 2D simplicial complexes, advancing the integration of harmonic map theory into their Teichmüller space analysis.

Contribution

It introduces foundational harmonic map results for ideal hyperbolic complexes, linking harmonic map theory with their Teichmüller spaces.

Findings

Existence of harmonic maps between the complexes.

Regularity properties of these harmonic maps.

Connection to Teichmüller space structures.

Abstract

We prove existence and regularity results for energy minimizing maps between ideal hyperbolic 2-dimensional simplicial complexes. The spaces in question were introduced by Charitos-Papadopoulos, who describe their Teichm\"uller spaces and some compactifications. This work is a first step in introducing harmonic map theory into the Teichm\"uller theory of these spaces.