Gray, Jeremy J., Poincaré's uniformisation theorem

Based on an analysis of original printed works, this essay is entirely devoted to the internal history of Poincaré’s uniformisation theorem, which is closely related to the Riemann mapping theorem and one of the crucial points under discussion between the last decades of the Nineteenth century and the first two of the Twentieth in the period of arrangement of complex function theory according to modern views. The theorem was originally formulated (1883) as follows: if f(z, w) = 0 is a many-valued function of z, there is a uniformising parametrization z = z(t), w = w(t) such that f(z(t),w(t)) = 0 and the domain of the uniformising functions is, simple cases aside, conformally equivalent to a disc. The paper connects and expands on some parts of the recently published volume: U. Bottazioni, J. Gray, Hidden Harmony – Geometric Fantasies. The Rise of Complex Function Theory (Springer, 2013) see in particular §§ 7.7.2, 7.7.3, 8.3.2.