北京雁栖湖应用数学研究院

河井公大朗 , 钱超

Qiongling Li

2023年03月29日 15:20 至 16:20

1110

Zoom 928 682 9093 (BIMSA)

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 $\overrightarrow{q}=(q_2,\cdots ,q_n)$ on $X$, one can define a Higgs bundle $(\text{hyper}{k}_{X,n},\theta (\overrightarrow{q}))$ in the Hitchin section. We show there exists a harmonic metric $h$ on $(\text{hyper}{k}_{X,n},\theta (\overrightarrow{q}))$ satisfying (i) $h$ weakly dominates $h_X$; (ii) $h$ is compatible with the real structure. Here $h_X$ is the Hermitian metric on $\text{hyper}{k}_{X,n}$ 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 $(\text{hyper}{k}_{X,n},\theta (\overrightarrow{q}))$ satisfying (i)(ii) uniquely exists. This is joint work with Takuro Mochizuki.