Complete harmonic metrics and subharmonic functions on the unit disc

Let X be a Riemann surface, KX → X the canonical bundle, and TX→ X the dual bundle of the canonical bundle. For each integer r ≥ 2, each q ∈ H⁰(KXʳ), and each choice of the square root KX1/2 of the canonical bundle, we obtain a Higgs bundle (Kᵣ,Φ(q)), which is called a cyclic Higgs bundle. A diagonal harmonic metric h = (h₁, , hᵣ) on a cyclic Higgs bundle yields $r-1$-Hermitian metrics H₁, , Hᵣ₋₁ on TX, while h₁, hᵣ, and q yield a degenerate Hermitian metric Hᵣ on TX. A diagonal harmonic metric is said to be complete if the Kähler metrics induced by H₁,, Hᵣ₋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 (Kᵣ,Φ(q)) with a fixed determinant. The holomorphic section q induces a subharmonic weight function ϕq=1/rlog|q|² on KX, and a diagonal harmonic metric depends solely on this weight function ϕq. In this paper, we extend the uniqueness part of the theorem of Li-Mochizuki to any subharmonic weight function $φ$ whose exponential is C² outside a compact subset K ⊆ X. We also show that on the unit disc, a complete Hermitian metric associated with $φ$ always exists. Furthermore, on the unit disc, when $φ$ can be monotonically approximated by a family of weight functions (φ_ε)0 < ε< 1, where each φ_ε is smooth and defined on a disc D_ε= ∈ C |z| < 1 - ε\, we show that the corresponding family of complete metrics (h_ε)0 < ε< 1 converges monotonically to a complete metric h associated with $φ$ as ε 0.