Complete harmonic metrics and subharmonic functions on the unit disc

Abstract

Let $X$ be a Riemann surface with canonical bundle $K_X$ and holomorphic tangent bundle $T_X=K_X^{-1}$. For each integer $r\ge 2$, each $q\in H^0(K_X^r)$, and each choice of a square root of $K_X$, we obtain a cyclic Higgs bundle. A diagonal harmonic metric $h=(h_1,\dots ,h_r)$ induces Hermitian metrics $H_j=h_j^{-1}\otimes h_{j+1}$ on $T_X$ for $j=1,\dots ,r-1$, together with a possibly degenerate Hermitian metric $H_r$ determined by $h_1,h_r$, and $q$. We call $h$ complete if the Kähler metrics induced by $H_1,\dots ,H_{r-1}$ are all complete. Li--Mochizuki proved that if $q$ is nonzero unless $X$ is hyperbolic, there exists a unique complete harmonic metric on the corresponding cyclic Higgs bundle satisfying $\det (h)=1$. The Hitchin equation depends on $q$ only through the subharmonic weight ${\varphi }_q=\frac{1}{r}\log |q{|}^2$ on $K_X$. We establish uniqueness of complete solutions with determinant $1$ for general subharmonic weight functions $\phi$ whose exponentials are $C^2$ outside a compact subset of $X$. On the unit disc $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$, we also prove that a real complete solution exists for every subharmonic weight function. We further establish a monotone approximation theorem. If ${\phi }_{\epsilon }$ are smooth subharmonic weight functions on the discs ${\mathbb{D}}_{\epsilon }=\{z\in \mathbb{C}:|z|<1-\epsilon \}$ decreasing to $\phi$ as $\epsilon \searrow 0$, the corresponding complete solutions $h_{\epsilon }$, normalized by $\det (h_{\epsilon })=1$, converge monotonically to a complete solution $h$ associated with $\phi$.

arXiv commentv2: Some errors and typos are corrected. 24pages