标题： 循环Higgs丛上的调和度量的香农熵 II 显示英文标题

标题： Shannon entropy for harmonic metrics on cyclic Higgs bundles II

摘要： 设$X$为一个黎曼曲面，$K_X \rightarrow X$为典范丛。对于每个整数$r \geq 2$，每个$q \in H^0(K_X^r)$，以及每个典范丛的平方根$K_X^{1/2}$的选择，我们得到一个Higgs丛，称为循环Higgs丛。 一个循环Higgs丛上的对角调和度量$h = (h_1, \dots, h_r)$在$K_X^{-1} \rightarrow X$上产生$r-1$-埃尔米特度量$H_1, \dots, H_{r-1}$，而$h_1$，$h_r$和$q$在$K_X^{-1}\rightarrow X$上产生一个退化的埃尔米特度量$H_r$。 该 $r$-微分 $q$在 $K_X$上诱导了一个次调和权函数 $\phi_q=\frac{1}{r}\log|q|$，并且对角线调和度量仅取决于这个权函数。 在之前的论文中，作者研究了与任意次调和权函数 $\varphi$相关的调和度量的扩展，这同时也构造了 $K_X^{-1}\rightarrow X$上的厄米特度量 $H_1,\dots, H_r$。 特别地，作者引入了一个称为熵的函数，该函数量化了度量之间的相互不对齐程度$H_1,\dots, H_r$。在本文中，通过类比统计力学中的规范系综，我们进一步引入了一个称为自由能的量。当$H_1,\dots, H_{r-1}$均为完备且满足关于其逼近的条件时，我们给出了自由能在每一点减少的充分条件，当$r=2,3$时，我们也给出了熵在每一点增加的充分条件。此外，在单位圆盘$\mathbb{D}$上，当在紧子集外是$C^2$时，我们从熵和自由能的角度，给出了函数$e^\varphi h_\ast^{-1} \otimes h_{\mathbb{D}}$有界的必要充分条件，其中$h_{\mathbb{D}}$表示由庞加莱度量诱导的厄米特度量。这个结果扩展了Wan、Benoist-Hulin、Labourie-Toulisse和Dai-Li的工作。 显示英文摘要

摘要： 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.