Conformal structure of regions of the complex plane and the ring of holomorphic functions How is the conformal structure of regions of the complex plane determined by the integral domain of holomorphic functions defined on those regions?
Thanks

The conformal structure on the plane domain is completely determined by the ring of holomorphic functions. More precisely if two such rings are isomorphic, and isomorphism is identity on the constants, the regions are conformally equivalent.
If the rings are only isomorphic as abstract rings, then the regions are either
conformally equivalent or anti-conformally equivalent. 
This is due to Bers (BAMS 54, 1948) for plane domains and to Rudin (BAMS 61, 1955)
for open Riemann surfaces.
The idea is to consider the space of maximal ideals of the ring. They are in 1-1
correspondence with the points of the region.  
Remark. BAMS is free onine.
