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

TL;DR

This paper proves the existence and uniqueness of harmonic metrics on Higgs bundles over non-compact hyperbolic surfaces, extending results to specific real Lie group cases under boundedness conditions.

Contribution

It establishes existence and uniqueness of harmonic metrics on Hitchin section Higgs bundles over non-compact hyperbolic surfaces, including special cases for $SO(n,n+1)$ and $Sp(4,\mathbb R)$.

Findings

Existence of harmonic metrics satisfying dominance and compatibility conditions.

Uniqueness of harmonic metrics when differentials are bounded.

Extension of techniques to specific real Lie group Higgs bundles.

Abstract

Let $X$ be an arbitrary non-compact hyperbolic Riemann surface, that is, not $\mathbb C$ or $\mathbb C^*$. Given a tuple of holomorphic differentials $\boldsymbol q=(q_2,\cdots,q_n)$ on $X$, one can define a Higgs bundle $(\mathbb{K}_{X,n},\theta(\boldsymbol q))$ in the Hitchin section. We show there exists a harmonic metric $h$ on $(\mathbb{K}_{X,n},\theta(\boldsymbol q))$ 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 $(\mathbb{K}_{X,n},\theta(\boldsymbol q))$ 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.