Discrete harmonic maps between hyperbolic surfaces

TL;DR

This paper introduces a method to find discrete harmonic maps from a topologically decomposed surface into hyperbolic surfaces by minimizing Dirichlet energy, revealing a connection to weighted Delaunay decompositions.

Contribution

It provides a novel variational approach to characterize discrete harmonic maps into hyperbolic surfaces using energy minimization and Delaunay structures.

Findings

Optimal hyperbolic structures induce discrete harmonic maps.

Edge weights correspond to weighted Delaunay decompositions.

Minimization yields a unique extremum characterized by geometric conditions.

Abstract

Given a topological cell decomposition of a closed surface equipped with edge weights, we consider the Dirichlet energy of any geodesic realization of the 1-skeleton graph to a hyperbolic surface. By minimizing the energy over all possible hyperbolic structures and over all realizations within a fixed homotopy class, one obtains a discrete harmonic map into an optimal hyperbolic surface. We characterize the extremum by showing that at the optimal hyperbolic structure, the discrete harmonic map and the edge weights are induced from a weighted Delaunay decomposition.