Complete harmonic metrics and subharmonic functions on the unit disc

TL;DR

This paper extends the uniqueness of complete harmonic metrics on cyclic Higgs bundles over Riemann surfaces to broader subharmonic weight functions, and constructs such metrics explicitly on the unit disc, including approximation methods.

Contribution

It generalizes Li-Mochizuki's uniqueness theorem to subharmonic weights with certain regularity and provides explicit constructions and approximation techniques for complete metrics on the unit disc.

Findings

Extended uniqueness of harmonic metrics to broader subharmonic weights.

Constructed complete Hermitian metrics on the unit disc for general subharmonic weights.

Established convergence of approximate metrics to the exact complete metric.

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 $\epsilon\searrow 0$.