Shannon entropy for harmonic metrics on cyclic Higgs bundles

Let $X$ be a Riemann surface, $K_X \rightarrow X$ the canonical bundle, and $T_X= K_X^{-1}\rightarrow X$ the dual bundle of 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 canonically 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 $T_X\rightarrow X$, defined as $H_j=h_j^{-1} \otimes h_{j+1}$ for each $j=1,\dots, r-1$, while $h_1$, $h_r$, and $q$ yield a degenerate Hermitian metric $H_r$ on $T_X \rightarrow X$. The $r$-differential $q$ induces a subharmonic weight function $\phi_q=\frac{1}{r}\log|q|^2$ on $K_X\rightarrow X$, and the diagonal harmonic metric depends solely on this weight function $\phi_q$. In the previous papers, the author introduced and studied the extension of harmonic metrics associated with arbitrary subharmonic weight function $\varphi$, which also constructs $r-1$-Hermitian metrics $H_1,\dots, H_{r-1}$ and a degenerate Hermitian metric $H_r$ on $T_X\rightarrow X$. In this paper, for each non-zero real parameter $\beta$, we introduce a function, which we call entropy, that quantifies the degree of mutual misalignment of the Hermitian metrics $H_1,\dots, H_r$. 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 $H_1,\dots, H_{r-1}$ 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 $\beta>-1$.