Mathematics

The uniformization theorem states that every Riemann surface can be written as quotient of one of the three simply connected Riemann surfaces, i.e. the complex sphere, the complex plane and the upper half plane, over a group of Möbius transformations G.  Despite this theorem was proved a long time ago, we are not able to find the group G in the general case. The main difficulty of this problem is to find quantities called accessory parameters. As a matter of fact if those parameters are known, the problem reduces to calculate the monodromy of a differential equation.
In this talk I will show how to construct the uniformization of a sphere with three missing points. Moreover I will present a result of Zograf and Takhtajan, which show that the accessory parameters in the case of the sphere with n-missing points are the partial derivatives of a functional evaluated on the solution of the so called Liouville equation, where this evaluation is seen as a function of the missing points w_1..w_n.