Shannon entropy for harmonic metrics on cyclic Higgs bundles II

TL;DR

This paper introduces the concepts of free energy and entropy in the context of harmonic metrics on cyclic Higgs bundles, providing conditions for their behavior and extending previous work on subharmonic weight functions on Riemann surfaces.

Contribution

It defines free energy for harmonic metrics on cyclic Higgs bundles and establishes conditions for its decrease and entropy increase, extending prior research on subharmonic weights.

Findings

Sufficient conditions for free energy to decrease at each point.

Conditions for entropy to increase when r=2,3.

Necessary and sufficient conditions for boundedness of a specific metric on the unit disc.

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.