Uniformizing surfaces via discrete harmonic maps

2019/05/14 by Toru Kajigaya, Ryokichi Tanaka, Kajigaya, Toru +1 · 2 citations
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #math.DG #math.MG

paper · pdf · doi:10.48550/arxiv.1905.05427

31 pages, 5 figures

openalex publication_date 2019/05/14 · arxiv created 2020/07/24 · arxiv updated 2020/07/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that for any closed surface of genus greater than one and for any finite weighted graph filling the surface, there exists a hyperbolic metric which realizes the least Dirichlet energy harmonic embedding of the graph among a fixed homotopy class and all hyperbolic metrics on the surface. We give explicit examples of such hyperbolic surfaces through a new interpretation of the Nielsen realization problem for the mapping class groups.

Citations

• Curves on 2-manifolds and isotopies

Cited by

• A Structure-Preserving Numerical Method for Harmonic Maps Between High-genus Surfaces
• Computing Harmonic Maps and Conformal Maps on Point Clouds

Related

• Discrete harmonic maps between hyperbolic surfaces
• A Structure-Preserving Numerical Method for Harmonic Maps Between High-genus Surfaces
• Minimal graphs over Riemannian surfaces and harmonic diffeomorphisms
• Minimal surfaces and harmonic diffeomorphisms from the complex plane onto a Hadamard surface
• Quadratic differentials, half-plane structures, and harmonic maps to graphs