A factorization theorem for harmonic maps

TL;DR

This paper proves a factorization theorem for harmonic maps from Riemann surfaces, showing how symmetries induce a factorization through a holomorphic map, extending known results for minimal maps.

Contribution

It establishes a new factorization result for harmonic maps based on their symmetries, generalizing classical minimal surface theory using properties of the Hopf differential.

Findings

Harmonic maps with certain symmetries factor through holomorphic maps.

Extension of minimal surface results to more general harmonic maps.

Use of Hopf differential properties in the proof.

Abstract

Let $f$ be a harmonic map from a Riemann surface to a Riemannian $n$-manifold. We prove that if there is a holomorphic diffeomorphism $h$ between open subsets of the surface such that $f\circ h = f$, then $f$ factors through a holomorphic map onto another Riemann surface. If such $h$ is anti-holomorphic, we obtain an analogous statement. For minimal maps, this result is well known and is a consequence of the theory of branched immersions of surfaces due to Gulliver-Osserman-Royden. Our proof relies on various geometric properties of the Hopf differential.