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 \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
where
U is a Jordan domain in
\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
\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
\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
\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
\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
\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
\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
\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.
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