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

Abstract

Let $X$ be an arbitrary non-compact hyperbolic Riemann surface, that is, not $\mathbb{C}$ or ${\mathbb{C}}^{\ast }$. Given a tuple of holomorphic differentials $\mathbf{q}=(q_2,\cdots ,q_n)$ on $X$, one can define a Higgs bundle $({\mathbb{K}}_{X,n},\theta (\mathbf{q}))$ in the Hitchin section. We show there exists a harmonic metric $h$ on $({\mathbb{K}}_{X,n},\theta (\mathbf{q}))$ satisfying (i) $h$ weakly dominates $h_X$; (ii) $h$ is compatible with the real structure. Here $h_X$ is the Hermitian metric on ${\mathbb{K}}_{X,n}$ induced by the conformal complete hyperbolic metric $g_X$ on $X.$ Moreover, when $q_i(i=2,\cdots ,n)$ are bounded with respect to $g_X$, we show such a harmonic metric on $({\mathbb{K}}_{X,n},\theta (\mathbf{q}))$ satisfying (i)(ii) uniquely exists. With similar techniques, we show the existence of harmonic metrics for $SO(n,n+1)$-Higgs bundles in Collier's component and $Sp(4,\mathbb{R})$-Higgs bundles in Gothen's component over $X$, under some mild assumptions.

38 pages, comments are very welcome