Harmonic metrics of $\mathrm{SO}_{0}(n,n)$-Higgs bundles in the Hitchin section on non-compact hyperbolic surfaces

Abstract

Let $X$ be a Riemann surface. Hitchin constructed the $G$-Higgs bundles in the Hitchin section for a split real form $G$ of a complex simple Lie group,using the canonical line bundle $K$ and some holomorphic differentials $\boldsymbol{q}$. We study the case of ${\mathrm{SO}_0(n,n)}$. In our work, we establish the existence of harmonic metrics for these Higgs bundles, which are compatible with the ${\mathrm{SO}_0(n,n)}$-structure on any non-compact hyperbolic Riemann surface. Furthermore, these harmonic metrics weakly dominate $h_X$, the natural diagonal harmonic metric induced by the unique complete K\"ahler hyperbolic metric $g_X$ on $X$. Assuming these holomorphic differentials are all bounded with respect to $g_X$, we prove the uniqueness of such a harmonic metric.