In his pioneering paper *An extension of Schwarz's lemma*, Ahlfors proves the lower bound on the Bloch constant $B \geq \frac{\sqrt{3}}{4}$. The proof of this lower bound proceeds as follows:


Let $W$ be a Riemann surface. For an arbitrary point $\mathfrak{m}$ of $W$, let $\rho(\mathfrak{m})$ denote the radius of the largest simple circle of centre $\mathfrak{m}$ contained in $W$. It is clear that $\rho(\mathfrak{m})$ is continuous, and is zero only at the branch points. We introduce the metric $ds = \lambda | dw |$, with $$\lambda = \frac{A}{2 \sqrt{\rho} (A^2-\rho)},$$ where $\rho = \rho(\mathfrak{m})$ and $w$ denotes the variable on the function plane (not the uniformizing variable), and $A^2>B(f)$ (the Bloch constant of $f$).


If $\mathfrak{a}$ is a branch point, then $\rho = | w - \mathfrak{a} |$. Let $n$ be the multiplicity of $\mathfrak{a}$. Then $w\_1 = (w-\mathfrak{a})^{\frac{1}{n}}$ is a uniformizing variable the corresponding $\lambda\_1$ is determined from $\lambda\_1 | dw\_1 | = \lambda | dw |$. Explicitly, $$\lambda\_1 = n \rho^{\frac{1}{2} - \frac{1}{n}} / 2(A^2 - \rho).$$ For $n=2$ the metric is regular, and for $n>2$, the metric is zero.



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> Ahlfors gives no explanation for why the $\lambda$ above is chosen. Is it simply constructed so that, with respect to the uniformizing variable, we obtain a continuous pseudo metric? Can one choose another metric to start with?
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Ahlfors' proof of Bloch's theorem
He explains his choice in lines 8-9 on p. 364 of the paper:


"This metric has the curvature $-4$ for it is obtained from the hyperbolic metric by the transformation $w'=w^{1/2}$ ."


Remarks. By more sophisticated choices of metrics later authors,
M. Bonk (MR0979048) and Chen and Gauthier (MR1428103) were able to improve Ahlfors's estimate of the Bloch constant.


