2025/08/18 by Natsuo Miyatake, Miyatake, Natsuo · 1 citation
Mathematics · #30C15 #31A05 #53C07 #Algebraic and Geometric Analysis #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2508.12848
openalex publication_date 2025/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (\mathbbKr,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 (\mathbbKr,Φ(q)) with a fixed determinant. The holomorphic section q induces a subharmonic weight function ϕq=(1)/(r)log|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 (φε)0 < ε< 1, where each φε is smooth and defined on a disc \mathbbDε= \z ∈ ℂ | |z| < 1 - ε\, we show that the corresponding family of complete metrics (hε)0 < ε< 1 converges monotonically to a complete metric h associated with φ as ε\searrow 0.