117 Guang-Yuan Zhang

Guang-Yuan Zhang (Tsinghua University, China) The precise form of Ahlfors' Second Fundamental Theorem of covering surfaces Abstract A simply connected covering surface $\Sigma =\left( f,\overline{\Delta }% \right) $ over the unit Riemann sphere $S$ is an orientation-preserving, continuous, open and finite-to-one mapping (OPCOFOM) $f$ from the closed unit disk $\overline{\Delta }$ into the sphere $S$. Here open means that $f$ can be extended continuous and open to a neighborhood of $\overline{\Delta }. $ We denote by $\mathbf{F}$ all simply connected surfaces. Let $E_{q}=\left\{ a_{1},a_{2},\dots ,a_{q}\right\} $ be a set on the unit Riemann sphere consisting of $q$ distinct points with $q>2.$ Ahlfors' second fundamental theorem (SFT) states that there exists a positive number $h$ depending only on $E_{q},$ such that for any surface $\Sigma =\left( f,% \overline{\Delta }\right) \in \mathbf{F},$ \[ \left( q-2\right) A\left( \Sigma \right) <4\pi \overline{n}\left( \Sigma \right) +hL\