Harmonic Metrics for Higgs Bundles of Rank 3 in the Hitchin Section

Given a tuple of holomorphic differentials on a Riemann surface, one can define a Higgs bundle in the Hitchin section and a natural symmetric pairing of the Higgs bundle. We study whether a Higgs bundle of rank 3 in the Hitchin section has a compatible harmonic metric when the spectral curve is a 2-sheeted branched covering of the Riemann surface. In particular, we give a condition for Higgs bundles in the Hitchin section on $\mathbb{C}$ or ${\mathbb{C}}^{\ast }$ to have compatible harmonic metrics.