Estimates of K\"ahler metrics on noncompact finite volume hyperbolic Riemann surfaces, and their symmetric products

TL;DR

This paper provides estimates for Kähler metrics on noncompact hyperbolic Riemann surfaces with finite volume and their symmetric products, advancing understanding of their geometric structures and metric properties.

Contribution

It derives explicit estimates for Bergman and Fubini-Study metrics on these surfaces and their symmetric products, extending previous work to noncompact finite volume cases.

Findings

Derived estimates for Bergman metrics on noncompact hyperbolic surfaces.

Established bounds for Fubini-Study metrics on symmetric products.

Connected metric estimates to volume forms on symmetric products.

Abstract

Let $X$ denote a noncompact finite volume hyperbolic Riemann surface of genus $g\geq 2$, with only one puncture at $i\infty$ (identifying $X$ with its universal cover $\mathbb{H}$). Let $\overline{X}:=X\cup\lbrace i\infty\rbrace$ denote the Satake compactification of $X$. Let $\Omega_{\overline{X}}$ denote the cotangent bundle on $\overline{X}$. For $k\gg1$, we derive an estimate for $\mu_{\overline{X}}^{\mathrm{Ber},k}$, the Bergman metric associated to the line bundle $\mathcal{L}^{k}:=\Omega_{\overline{X}}\otimes \mathcal{O}_{\overline{X}}\big((k-1)\infty\big)$. For a given $d\geq 1$, the pull-back of the Fubini-Study metric on the Grassmannian, which we denote by $\mu_{\mathrm{Sym}^d(\overline{X})}^{\mathrm{FS},k}$, defines a K\"ahler metric on $\mathrm{Sym}^d(\overline{X})$, the $d$-fold symmetric product of $\overline{X}$. Using our estimates of $\mu_{\overline{X}}^{\mathrm{Ber},k}$, as an application, we derive an estimate for $\mu_{\mathrm{Sym}^d(\overline{X}),\mathrm{vol}}^{\mathrm{FS},k}$, the volume form associated to the (1,1)-form $\mu_{\mathrm{Sym}^d(\overline{X})}^{\mathrm{FS},k}$.