Shannon entropy for harmonic metrics on cyclic Higgs bundles II

Let X be a Riemann surface and KX → X 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, 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 KX⁻¹ → X, while h₁, hᵣ, and q yield a degenerate Hermitian metric Hᵣ on KX⁻¹→ X. The r-differential q induces a subharmonic weight function ϕq=1/rlog|q| on KX, and the diagonal harmonic metric depends solely on this weight function. In the previous papers, the author studied the extension of harmonic metrics associated with arbitrary subharmonic weight function $φ$, which also constructs Hermitian metrics H₁,, Hᵣ on KX⁻¹→ X. Especially, the author introduced a function called entropy that quantifies the degree of mutual misalignment of the metrics H₁,, Hᵣ. In this paper, by analogy with the canonical ensemble in statistical mechanics, we further introduce the quantity which we call free energy. When H₁,, Hᵣ₋₁ are all complete and satisfy a condition concerning their approximation, we give a sufficient condition for the free energy to decrease at each point, and when $r=2,3$ we also give a sufficient condition for the entropy to increase at each point. Furthermore, on the unit disc D, when is C² outside a compact subset, we provide, from the perspective of entropy and free energy, necessary and sufficient conditions for the function e^φh_⁻¹ ⊗ hD to be bounded, where hD denotes the Hermitian metric induced by the Poincaré metric. This result extends the work of Wan, Benoist-Hulin, Labourie-Toulisse, and Dai-Li.