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

Time: 15:20-16:50, Mar. 29, 2023

Venue: BIMSA 1110

Zoom:  928 682 9093     Passcode: BIMSA

Abtract：

Let $X$ be an arbitrary non-compact hyperbolic Riemann surface, that is, not $\mathbb C$ or $\mathbb C^*$. Given a tuple of holomorphic differentials $\vecq=(q_2,\cdots,q_n)$ on $X$, one can define a Higgs bundle $(\hyperk_{X,n},\theta(\vecq))$ in the Hitchin section. We show there exists a harmonic metric $h$ on $(\hyperk_{X,n},\theta(\vecq))$ satisfying (i) $h$ weakly dominates $h_X$; (ii) $h$ is compatible with the real structure. Here $h_X$ is the Hermitian metric on $\hyperk_{X,n}$ 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 $(\hyperk_{X,n},\theta(\vecq))$ satisfying (i)(ii) uniquely exists. This is joint work with Takuro Mochizuki.