Shannon entropy for harmonic metrics on cyclic Higgs bundles II

Abstract

Let $X$ be a Riemann surface and $K_X \rightarrow X$ the canonical bundle. For each integer $r \geq 2$, each $q \in H^0(K_X^r)$, and each choice of the square root $K_X^{1/2}$ of the canonical bundle, we obtain a Higgs bundle, which is called a cyclic Higgs bundle. A diagonal harmonic metric $h = (h_1, \dots, h_r)$ on a cyclic Higgs bundle yields $r-1$-Hermitian metrics $H_1, \dots, H_{r-1}$ on $K_X^{-1} \rightarrow X$, while $h_1$, $h_r$, and $q$ yield a degenerate Hermitian metric $H_r$ on $K_X^{-1}\rightarrow X$. The $r$-differential $q$ induces a subharmonic weight function $\phi_q=\frac{1}{r}\log|q|$ on $K_X$, 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 $\varphi$, which also constructs Hermitian metrics $H_1,\dots, H_r$ on $K_X^{-1}\rightarrow X$. Especially, the author introduced a function called entropy that quantifies the degree of mutual misalignment of the metrics $H_1,\dots, H_r$. In this paper, by analogy with the canonical ensemble in statistical mechanics, we further introduce the quantity which we call free energy. When $H_1,\dots, H_{r-1}$ 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 $\mathbb{D}$, when is $C^2$ outside a compact subset, we provide, from the perspective of entropy and free energy, necessary and sufficient conditions for the function $e^\varphi h_\ast^{-1} \otimes h_{\mathbb{D}}$ to be bounded, where $h_{\mathbb{D}}$ denotes the Hermitian metric induced by the Poincar\'e metric. This result extends the work of Wan, Benoist-Hulin, Labourie-Toulisse, and Dai-Li.