A Note on harmonic maps

TL;DR

This paper clarifies and extends the representation of harmonic maps using Jacobi elliptic functions, building on recent and classical works in differential geometry.

Contribution

It provides detailed derivations and shows that harmonic maps from various studies can be expressed with Jacobi elliptic functions.

Findings

Harmonic maps from Shi, Tam, and Wan's work can be expressed with Jacobi elliptic functions.

Harmonic maps from Wang's work can also be represented using Jacobi elliptic functions.

The paper fills in details and unifies different approaches to harmonic maps through elliptic functions.

Abstract

In this note we will fill out the details from the recent work of Fotiadis and Daskaloyannis in arXiv:1903.05420v3, where the harmonic maps described by Y. Shi, L. Tam and T. Y.-H. Wan (in their work Harmonic Maps on Hyperbolic spaces with Singular Boundary Value, Differential Geometry 51(1999), 551-600) are written by the use of Jacobi elliptic functions. We also prove that the harmonic maps described by J. Wang in his work (The Heat Flow and Harmonic Maps between Complete Manifolds, The Gournal of Geometric Analysis, Volume 8, Number 3, 1998, 485-514) can be written by the use of Jacobi elliptic functions, too.