# complex geometry – Ahlfors’ proof of Bloch’s theorem

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.

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?