Higgs bundles in the Hitchin section over non‐compact hyperbolic surfaces

Abstract Let be an arbitrary non‐compact hyperbolic Riemann surface, that is, not the complex plane or punctured plane. Let be , or . We show that, on a Higgs bundle in the ‐Hitchin section over , there always exists a harmonic metric satisfying certain nice properties. Moreover, when the Higgs field has bounded spectral curve with respect to the complete hyperbolic metric on , such a harmonic metric is unique. As a result, we obtain a new proof of the existence and uniqueness of a harmonic metric for Higgs bundles in the ‐Hitchin section over a compact hyperbolic Riemann surface. Moreover, we show an existence result of harmonic metrics for a more general family of Higgs bundles that admit a full holomorphic filtration.