Ahlfors’ Second Fundamental Theorem for the simply connected surfaces over the Riemann sphere S states that for any set Eq of distinct q points on S with q ≥ 3, there exists a constant h = h(Eq), such that for any covering surface \(\Sigma=(f, \ {\overline U})\) , one has \((q-2) \ A(\Sigma) \leq 4{\pi}{\overline n}(\Sigma, \ E_{q})+hL(\partial\Sigma),\) where U is a Jordan domain in \({\mathbb C}, f : {\overline U} \rightarrow S\) 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 \({\overline n}(\Sigma, \ E_{q})=\#[f^{-1}(E_{q}) \ \cap U]\) means the cardinality of the set f−1 (Eq) ∩ U. We denote by F the space of all simply covering surfaces over the sphere, the above pairs \((f, \ {\overline U})\) , 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 \({\cal F}(L, \ m)\) of F (L), and Zhang asserted without proof in that paper that the extremal surface of \({\cal F}(L, \ m)\) can be found in the smaller subspace \({{\cal F}_{r}}(L, \ m)\) such that the defining function f of each surface of the form \((f, {\overline \Delta})\) in \({{\cal F}_{r}}(L, \ m)\) has no branch value outside Eq. In this paper, we prove this assertion.
