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.
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Acknowledgements
Many thanks to Vlad Markovic for encouragement and sharing helpful ideas. I would also like to thank John Wood and Jürgen Jost for comments on earlier drafts.
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Sagman, N. A Factorization Theorem for Harmonic Maps. J Geom Anal 31, 11714–11740 (2021). https://doi.org/10.1007/s12220-021-00699-w
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DOI: https://doi.org/10.1007/s12220-021-00699-w