Complete harmonic metrics and subharmonic functions on the unit disc

Abstract

Let $X$ be a Riemann surface, $K_X \rightarrow X$ the canonical bundle, and $T_X\rightarrow X$ the dual bundle of the canonical bundle. For each integer $r \geq 2$, each $q \in H^0(K_X^r)$, and each choice of the square root $K_X^{1/2}$ of the canonical bundle, we obtain a Higgs bundle $(\mathbb{K}_r,\Phi(q))$, which is called a cyclic Higgs bundle. A diagonal harmonic metric $h = (h_1, \dots, h_r)$ on a cyclic Higgs bundle yields $r-1$-Hermitian metrics $H_1, \dots, H_{r-1}$ on $T_X$, while $h_1$, $h_r$, and $q$ yield a degenerate Hermitian metric $H_r$ on $T_X$. A diagonal harmonic metric is said to be complete if the K\"ahler metrics induced by $H_1,\dots, H_{r-1}$ are all complete. Li-Mochizuki established a theorem stating that on any Riemann surface $X$ and any $q$ that is non-zero unless $X$ is hyperbolic, there exists a unique complete harmonic metric $h$ on $(\mathbb{K}_r,\Phi(q))$ with a fixed determinant. The holomorphic section $q$ induces a subharmonic weight function $\phi_q=\frac{1}{r}\log|q|^2$ on $K_X$, and a diagonal harmonic metric depends solely on this weight function $\phi_q$. In this paper, we extend the uniqueness part of the theorem of Li-Mochizuki to any subharmonic weight function $\varphi$ whose exponential is $C^2$ outside a compact subset $K \subseteq X$. We also show that on the unit disc, a complete Hermitian metric associated with $\varphi$ always exists. Furthermore, on the unit disc, when $\varphi$ can be monotonically approximated by a family of weight functions $(\varphi_\epsilon)_{0 < \epsilon < 1}$, where each $\varphi_\epsilon$ is smooth and defined on a disc $\mathbb{D}_\epsilon= \{z \in \mathbb{C} \mid |z| < 1 - \epsilon\}$, we show that the corresponding family of complete metrics $(h_\epsilon)_{0 < \epsilon < 1}$ converges monotonically to a complete metric $h$ associated with $\varphi$ as $\epsilon\searrow 0$.