Branch values in Ahlfors' theory of covering surfaces (Q2193954)

Paper's results help to determine two important constants appearing in Ahlfors' second fundamental theorem in the theory of covering surfaces. A domain \(U\) on the unit sphere \(S\subset\mathbb R^3\) or on the extended complex plane \(\overline{\mathbb C}\) is regular if \(\partial U\) is a finite union of disjoint Jordan curves. A \(\overline{\mathbb C}\)-valued function \(f\) on a (closed) domain \(W\) is discrete if, for each \(p\in S\), \(f^{-1}(p)\) is discrete in \(W\). A \(\overline{\mathbb C}\)-valued function \(\tilde f\) from a domain \(W\) on \(S\) is an orientation-preserving light mapping if \(\tilde f\) is continuous, open, orientation-preserving and discrete. Let \(\Sigma=(f,\overline U)\) be a covering surface for a regular domain \(U\) on \(S\) and an orientation-preserving light mapping \(f\) on the closure \(\overline U\) of \(U\). Denote by \(L(\partial\Sigma)=L(f,\partial U)\) and \(A(\Sigma)\) the spherical perimeter and the spherical area of \(\Sigma\), respectively. For \(w\in S\), let \(\overline n(f,w)=\overline n(\Sigma,w)\) be the covering number of \(f\) in \(U\). The family of all simply connected covering surfaces \(\Sigma\) is denoted by \(\mathbf F\). Ahlfors' second fundamental theorem states that, for any set \(\Sigma_q=\{a_1,\dots,a_q\}\) consisting of \(q\) distinct points on \(S\), \(q\geq3\), there exists a constant \(h>0\), dependent only on \(E_q\), such that, for each \(\Sigma\in\mathbf F\), \[(q-2)A(\Sigma)\leq4\pi\sum_{j=1}^q\overline n(\Sigma,a_j)+hL(\partial\Sigma).\] The total covering number \(\overline n(f,E_q)=\overline n(\Sigma,E_q)\) is defined as the sum \(\sum_{j=1}^q\overline n(\Sigma,a_j)\), the remainder \(R(\Sigma,E_q)\) is defined as \((q-2)A(\Sigma)-4\pi\overline n(\Sigma,E_q)\), and the remainder-perimeter ratio \(H(\Sigma,E_q)\) is defined as \(R(\Sigma,E_q)/L(\partial\Sigma)\). Denote \(\Delta=\{z\in\mathbb C:|z|<1\}\). The main result of the paper is given in the following theorem. Theorem 1.7. For each \(\Sigma_0\in\mathbf F\) there exists \(\Sigma_1=(f_1,\overline{\Delta})\in\mathbf F\) such that the following holds: (1) \(H(\Sigma_1,E_q)\geq H(\Sigma_0,E_q)\), \(\overline n(\Sigma_1,E_q)\leq\overline n(\Sigma_0,E_q)\) and \(L(\partial\Sigma_1)\leq L(\partial\Sigma_0)\). (2) \(f_1\) is locally homeomorphic near each \(z\in\overline{\Delta}\setminus f_1^{-1}(E_q)\). (3) If \(R(\Sigma_0,E_q)\geq0\), then there is a rotation \(\sigma\) of \(S\) such that \(\partial\Sigma_1\subset\sigma(\partial\Sigma_0)\).