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

组织者

河井公大朗 , 钱超

演讲者

Qiongling Li

时间

2023年03月29日 15:20 至 16:20

地点

1110

线上

Zoom 928 682 9093 (BIMSA)

摘要

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