2024/10/11 by Miyatake, Natsuo
#30C15 #31A05 #53C07 #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2410.08571
Let X be a Riemann surface, KX → X the canonical bundle, and TX= KX-1→ 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 canonically obtain a Higgs bundle, 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→ X, defined as Hj=hj-1 ⊗ hj+1 for each j=1,…, r-1, while h1, hr, and q yield a degenerate Hermitian metric Hr on TX → X. The r-differential q induces a subharmonic weight function ϕq=(1)/(r)log|q|2 on KX→ X, and the diagonal harmonic metric depends solely on this weight function ϕq. In the previous papers, the author introduced and studied the extension of harmonic metrics associated with arbitrary subharmonic weight function φ, which also constructs r-1-Hermitian metrics H1,…, Hr-1 and a degenerate Hermitian metric Hr on TX→ X. In this paper, for each non-zero real parameter β, we introduce a function, which we call entropy, that quantifies the degree of mutual misalignment of the Hermitian metrics H1,…, Hr. By extending the estimate established by Dai-Li and Li-Mochizuki to general subharmonic weight functions, we provide an upper bound and a lower bound for the entropy when H1,…, Hr-1 are all complete and satisfy a condition concerning their approximation. Additionally, we show that the difference between the lower and upper bounds of entropy converges to a finite real number if and only if β>-1.