Harmonic metrics of SO₀ (n, n) -Higgs bundles in the Hitchin section on non-compact hyperbolic surfaces

Let X be a Riemann surface. Using the canonical line bundle K and some holomorphic differentials q, Hitchin constructed the G-Higgs bundles in the Hitchin section for a split real form G of a complex simple Lie group. We study the SO₀ (n, n) case. In our work, we establish the existence of harmonic metrics for these Higgs bundles, which are compatible with the SO₀ (n, n) -structure for any non-compact hyperbolic Riemann surface. Moreover, these harmonic metrics also weakly dominate hX which is the natural diagonal harmonic metric induced by the unique complete K\"ahler hyperbolic metric gX on X. Assuming that these holomorphic differentials are all bounded with respect to the metric gX, we are able to prove the uniqueness of such a harmonic metric.