A Finite Theorem for Ahlfors’ Covering Surface Theory

Abstract

Ahlfors’ theory of covering surfaces is a great mathematical achievement of the last century. The most important part of this theory is the second fundamental theorem (SFT for short). The authors are interested in the relation between the error terms of Ahlfors’ SFT for surfaces with the same boundary curve.

In this paper, the authors prove a result that is used to establish the best bound of the constant in Ahlfors’ SFT (in 2023) for simply connected surfaces.

Specifically, the authors prove that for any surface Σ in the space

{\mathcal{F}}_r(L,m)={\mathcal{F}}_r(L,m,E_q)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal{F}}_{r}(L,m) = {\mathcal{F}}_{r}(L,m,E_{q})$$\end{document}

, a surface Σ′ in the subspace

{\mathcal{F}}_r^{\mathrm{\prime }}(L,m)={\mathcal{F}}_r^{\mathrm{\prime }}(L,m,E_q)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal{F}}_{r}^{\prime}(L,m) = {\mathcal{F}}_{r}^{\prime}(L,m,E_{q})$$\end{document}

of

{\mathcal{F}}_r(L,m)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal{F}}_{r}(L,m)$$\end{document}

can be constructed such that Σ′ and Σ share the same boundary and the same Ahlfors error term, namely, ∂Σ′ = ∂Σ and R(Σ′, E_q) = R(Σ, E_q). This is important in the study of the ratio

H(\mathrm{\Sigma })=\frac{R(\mathrm{\Sigma })}{L(\mathrm{\partial }\mathrm{\Sigma })}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H(\Sigma)=\frac{R(\Sigma)}{L(\partial\Sigma)}$$\end{document}

of the error term R(Σ) to the boundary length L(∂Σ), since it implies that

\underset{\mathrm{\Sigma }\in {\mathcal{F}}_r(L,m)}{sup}H(\mathrm{\Sigma })=\underset{{\mathrm{\Sigma }}^{\mathrm{\prime }}\in {\mathcal{F}}_r^{\prime }(L,m)}{sup}H({\mathrm{\Sigma }}^{\mathrm{\prime }}),

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sup_{\Sigma \in \mathcal{F}_{r}(L,m)} H(\Sigma) = \sup_{\Sigma^{\prime} \in \mathcal{F}_{r}'(L,m)} H(\Sigma^{\prime}),$$\end{document}

{\mathcal{F}}_r^{\mathrm{\prime }}(L,m)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal{F}}_{r}^{\prime}(L,m)$$\end{document}

is a simpler subspace of

{\mathcal{F}}_r(L,m)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal{F}}_{r}(L,m)$$\end{document}

, and

H_0=\underset{L\to \mathrm{\infty }}{lim}\underset{\mathrm{\Sigma }\in {\mathcal{F}}_r(L,m)}{sup}H(\mathrm{\Sigma })

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H_0=\lim_{L\to\infty}\sup_{\Sigma\in\mathcal{F}_r(L,m)}H(\Sigma)$$\end{document}

is the best bound of the constant in Ahlfors’ SFT among all simply connected surfaces.

Keywords

Ahlfors’ constant / Covering surface / Isoperimetric inequality / Value distribution theory / 30D35 / 30D45 / 52B60

References

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