Movement of Branch Points in Ahlfors’ Theory of Covering Surfaces

Abstract

Ahlfors’ Second Fundamental Theorem for the simply connected surfaces over the Riemann sphere S states that for any set E _q of distinct q points on S with q ≥ 3, there exists a constant h = h(E _q), such that for any covering surface

\mathrm{\Sigma } = ( f , \overline{U} )

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Sigma=(f, \ {\overline U})$$\end{document}

, one has

( q - 2 ) A ( \mathrm{\Sigma } ) \le 4 \pi \overline{n} ( \mathrm{\Sigma } , E_q ) + h 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}$$(q-2) \ A(\Sigma) \leq 4{\pi}{\overline n}(\Sigma, \ E_{q})+hL(\partial\Sigma),$$\end{document}

where U is a Jordan domain in

\mathbb{C} , f : \overline{U} \to S

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb C}, f : {\overline U} \rightarrow S$$\end{document}

is an orientation-preserving, continuous, open and finite-to-one mapping, A(Σ) is the spherical area of Σ, L(∂Σ) is the spherical length of the boundary of Σ and

\overline{n} ( \mathrm{\Sigma } , E_q ) = \mathrm{\#} [ f^{-1} ( E_q ) \cap U ]

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\overline n}(\Sigma, \ E_{q})=\#[f^{-1}(E_{q}) \ \cap U]$$\end{document}

means the cardinality of the set f⁻¹ (E _q) ∩ U. We denote by F the space of all simply covering surfaces over the sphere, the above pairs

( f , \overline{U} )

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(f, \ {\overline U})$$\end{document}

, and write F (L) = {Σ ∈ F: L(∂Σ) ≤ L}. In a preprint by the third author Zhang of this paper, the precised bound of h is identified (see arXiv.2307.04623). The first key step of Zhang’s method is to prove the existence of extremal surfaces of the subspace

\mathcal{F} ( L , m )

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

of F (L), and Zhang asserted without proof in that paper that the extremal surface of

\mathcal{F} ( L , m )

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

can be found in the smaller subspace

{\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}$${{\cal F}_{r}}(L, \ m)$$\end{document}

such that the defining function f of each surface of the form

( f , \overline{\mathrm{\Delta }} )

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(f, {\overline \Delta})$$\end{document}

in

{\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}$${{\cal F}_{r}}(L, \ m)$$\end{document}

has no branch value outside E _q. In this paper, we prove this assertion.

Keywords

Nevanlinna’s theory of value distribution / Ahlfors’ theory of covering surfaces / isoperimetric inequality / spherical geometry

References

[1] AhlforsL Zur Theorie der Überlagerungsflächen. Acta Math., 1935, 65(1): 157-194

[2] AhlforsL Complex Analysis—An Introduction to the Theory of Analytic Functions of One Complex Variable, 1978 Third New York McGraw-Hill Book Co.

[3] EremenkoA. Ahlfors’ contribution to the theory of meromorphic functions. Lectures in Memory of Lars Ahlfors (Haifa, 1996), 2000 Ramat Gan Bar-Ilan Univ. 41-63 14

[4] RickmanS Quasiregular Mappings, 1993 Berlin Springer-Verlag Results in Mathematics and Related Areas (3)] 26

[5] StoïlowS Leçons sur les principes topologiques de la théorie des fonctions analytiques, Deuxième édition, augmentée de notes sur les fonctions analytiques et leurs surfaces de Riemann, 1956 Paris Gauthier-Villars

[6] SunZ, ZhangG. Branch values in Ahlfors’ theory of covering surfaces. Sci. China Math., 2020, 63(8): 1535-1558

[7] ZhangG. The precise bound for the area-length ratio in Ahlfors’ theory of covering surfaces. Invent. Math., 2013, 191(1): 197-253

[8] Zhang G., The precise form of Ahlfors’ second fundamental theorem. 2023, arXiv:2307.04623v1

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