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 $\Sigma =\left( f,% \overline{\Delta }\right) \in \mathbf{F},$ \[ \left( q-2\right) A\left( \Sigma \right) <4\pi \overline{n}\left( \Sigma \right) +hL\left( \partial \Sigma \right) , \] where $\Delta $ is the unit disk, $A\left( \Sigma \right) $ is the spherical area of $\Sigma $, $L\left( \partial \Sigma \right) $ is the spherical length of the boundary curve $\partial \Sigma =\left( f,\partial \Delta \right) ,$ and $\overline{n}\left( \Sigma \right) =\#f^{-1}(E_{q})\cap \Delta .$

If we define $R\left( \Sigma \right) =R\left( \Sigma ,E_{q}\right) $ to be the error term in Ahlfors' SFT, say, \[ R\left( \Sigma \right) =\left( q-2\right) A\left( \Sigma \right) -4\pi \overline{n}\left( \Sigma \right) , \] then Ahlfors' SFT reads \[ H_{0}=\sup_{\Sigma \in \mathbf{F}}\left\{ \frac{R(\Sigma )}{L(\partial \Delta )}:\Sigma =\left( f,\overline{\Delta }\right) \right\} <+\infty . \] We call $H_{0}=H_{0}(E_{q})$ Ahlfors' constant for simply connected surfaces.

In this talk, I will introduce my recent work which identify the precise bound $H_{0}=H_{0}(E_{q}).$

complex variablesdynamical systems

Audience: researchers in the topic

CAvid: Complex Analysis video seminar

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