Shannon entropy for harmonic metrics on cyclic Higgs bundles

TL;DR

This paper introduces an entropy measure for harmonic metrics on cyclic Higgs bundles over Riemann surfaces, providing bounds and convergence criteria related to the metrics' mutual alignment and subharmonic weight functions.

Contribution

It extends previous work by defining a new entropy function for harmonic metrics parameterized by 2, and establishes bounds and convergence conditions for this entropy in the context of cyclic Higgs bundles.

Findings

Provides upper and lower bounds for the entropy in terms of harmonic metrics.

Shows the entropy difference converges to a finite value if and only if 2 > -1.

Extends estimates to general subharmonic weight functions.

Abstract

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 $\beta>-1$.