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.

中文翻译：

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调和图的因式分解定理

令f为从Riemann曲面到Riemannian n流形的调和图。我们证明，如果在表面的开放子集之间存在全纯微分h，使得\（f \ circ h = f \），则f通过全纯图映射到另一个Riemann曲面上。如果这样的h是反全纯的，我们得到一个类似的陈述。对于最小的地图，此结果是众所周知的，并且是Gulliver-Osserman-Royden归因于表面分支浸没理论的结果。我们的证明依赖于Hopf微分的各种几何特性。