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The precise bound for the area–length ratio in Ahlfors’ theory of covering surfaces

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Abstract

Let \(S=\overline{\mathbb{C}}\) be the unit Riemann sphere. A basic consequence of Ahlfors’ theory of covering surfaces is that there exists a positive constant h such that for any simply-connected surface Σ over S ∗=S∖{0,1,∞},

$$A(\varSigma)\leq hL(\partial\varSigma).$$

Here A(Σ) is the area of Σ and L(∂Σ) is the length of the boundary of Σ. The goal of this paper is to prove that the least possible value of h is

$$h_{0}=\max_{\theta\in\lbrack0,\pi/2]} \biggl[ \frac{ ( \pi+\theta ) \sqrt{1+\sin^{2}\theta}}{\arctan\frac{\sqrt{1+\sin^{2}\theta}}{\cos\theta}}-\sin\theta \biggr] =4.03415979051\ldots $$

We develop a new method that not only reinterprets Ahlfors’ inequality, but also gives the precise bound.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, P.R. China

    Guang Yuan Zhang

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  1. Guang Yuan Zhang
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Correspondence to Guang Yuan Zhang.

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To my father Lu Sheng Zhang (1935–2007)

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Zhang, G.Y. The precise bound for the area–length ratio in Ahlfors’ theory of covering surfaces. Invent. math. 191, 197–253 (2013). https://doi.org/10.1007/s00222-012-0398-z

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  • DOI: https://doi.org/10.1007/s00222-012-0398-z

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