The precise bound for the area–length ratio in Ahlfors’ theory of covering surfaces
The precise bound for the area–length ratio in Ahlfors’ theory of covering surfaces
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DOI：
10.1007/s00222-012-0398-z
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发表时间：
2012-04
影响因子：
3.1
通讯作者：
G. Zhang
中科院分区：
文献类型：
--
作者：
G. Zhang
Letbe the unit Riemann sphere. A basic consequence of Ahlfors’ theory of covering surfaces is that there exists a positive constanthsuch that for any simply-connected surfaceΣoverS∗=S∖{0,1,∞},HereA(Σ) is the area ofΣandL(∂Σ) is the length of the boundary ofΣ. The goal of this paper is to prove that the least possible value ofhis $$h_{0}=\max_{\theta\in\lbrack0,\pi/2]} \biggl[ \frac{ ( \pi+\theta ) \sqrt{1+\sin^{2}\theta}}{\arctan\frac{\sqrt{1+\sin^{2}\theta}}{\cos\theta}}-\sin\theta \biggr] =4.03415979051\ldots $$We develop a new method that not only reinterprets Ahlfors’ inequality, but also gives the precise bound.
Letbe the unit Riemann sphere. A basic consequence of Ahlfors’ theory of covering surfaces is that there exists a positive constanthsuch that for any simply-connected surfaceΣoverS∗=S∖{0,1,∞},HereA(Σ) is the area ofΣandL(∂Σ) is the length of the boundary ofΣ. The goal of this paper is to prove that the least possible value ofhis $$h_{0}=\max_{\theta\in\lbrack0,\pi/2]} \biggl[ \frac{ ( \pi+\theta ) \sqrt{1+\sin^{2}\theta}}{\arctan\frac{\sqrt{1+\sin^{2}\theta}}{\cos\theta}}-\sin\theta \biggr] =4.03415979051\ldots $$We develop a new method that not only reinterprets Ahlfors’ inequality, but also gives the precise bound.