Shannon entropy for harmonic metrics on cyclic Higgs bundles

Abstract

Let $X$ be a Riemann surface, $K_X$ its canonical bundle, and $T_X=K_X^{-1}$. A diagonal harmonic metric $h=(h_1,\dots ,h_r)$ on a cyclic Higgs bundle of rank $r\ge 2$ associated with a holomorphic $r$-differential $q$ induces Hermitian metrics $H_j=h_j^{-1}\otimes h_{j+1}$ on $T_X$ for $j=1,\dots ,r-1$, together with a possibly degenerate metric $H_r$ determined by $h_1,h_r$, and $q$. This construction depends on $q$ only through the subharmonic weight ${\varphi }_q=\frac{1}{r}\log |q{|}^2$ on $K_X$. In previous work, we extended harmonic metrics to solutions associated with arbitrary subharmonic weight functions $\phi$, yielding analogous metrics $H_1,\dots ,H_r$. For each nonzero real parameter $\beta$, we introduce a function on $X$, which we call entropy, that quantifies their mutual misalignment. By extending estimates of Dai--Li and Li--Mochizuki to general subharmonic weight functions, we obtain upper and lower bounds for $\beta >0$ when $H_1,\dots ,H_{r-1}$ are complete and the weight is neither induced by a complete flat Kähler metric nor identically $-\infty$. The upper bound is $\log r$, the constant entropy associated with a complete flat Kähler metric, for which $H_1=\cdots =H_r$. The lower bound is the constant entropy associated with the weight $\phi \equiv -\infty$. Furthermore, for every nonzero real $\beta$, we determine the asymptotic behavior, as $r\to \infty$, of the difference between these two entropy values, showing that it converges to a finite real number if and only if $\beta >-1$.

arXiv commentv3: The approximation assumption in Theorem 1 has been removed, and the proof of Theorem 2 has been corrected. 29 pages