an:06143189
Zbl 1281.30021
Zhang, Guang Yuan
The precise bound for the area-length ratio in Ahlfors' theory of covering surfaces
EN
Invent. Math. 191, No. 1, 197-253 (2013).
00314112
2013
j
30D20 30D35 30D45 51M25
covering surface; value distribution; isoperimetric inequality; spherical geometry
A key result in Ahlfors' theory of covering surfaces is that for any surface \(\Sigma\) over \(\overline{\mathbb{C}} \setminus \{ 0,1,\infty \}\), there exists a constant \(h\) such that the isoperimetric inequality
\[
A(\Sigma) \leq h L(\partial \Sigma)
\]
holds, where \(A,L\) denote area and length weighted according to multiplicity, respectively. This follows from an interpretation of the Second Fundamental Theorem of Nevanlinna Theory, and has further important consequences for complex analysis.
In the paper under review, the author obtains the smallest possible value of \(h\), given by
\[
h_0 = \max_{\theta \in [0,\pi/2]} \frac{ (\pi + \theta )\sqrt{1+\sin^2\theta}}{\arctan \left ( \frac{ \sqrt{1+\sin^2\theta}}{\cos\theta} \right ) }.
\]
This is shown to be sharp by explicit construction of surfaces \(\Sigma_n\) with
\[
\lim_{n\to \infty} \frac{A(\Sigma_n)}{L(\partial \Sigma_n)} = h_0.
\]
In this well-written paper, the author proceeds by establishing the inequality for a certain family of surfaces \(\mathbb{F}^*\), and then showing that this family well-approximates any other such surface in the sense of the quotient \(A/L\).
Alastair Fletcher (Dekalb)