Computing harmonic maps between Riemannian manifolds

TL;DR

This paper investigates how discrete harmonic maps between Riemannian manifolds, approximated via weighted triangulations, converge to smooth harmonic maps, supported by theoretical conditions and computational implementation.

Contribution

It establishes conditions for convergence of discrete harmonic maps to smooth ones and introduces software for computing equivariant harmonic maps in hyperbolic space.

Findings

Conditions ensuring convergence of discrete to smooth harmonic maps

Implementation of software Harmony for hyperbolic plane maps

Validation of convergence through computational experiments

Abstract

In the previous paper [GLM2018], we showed that the theory of harmonic maps between Riemannian manifolds may be discretized by introducing triangulations with vertex and edge weights on the domain manifold. In the present paper, we study convergence of the discrete theory to the smooth theory when taking finer and finer triangulations. We present suitable conditions on the weighted triangulations that ensure convergence of discrete harmonic maps to smooth harmonic maps. Our computer software Harmony implements these methods to computes equivariant harmonic maps in the hyperbolic plane.