I'm looking for a reference for the following fact: given two Riemann surfaces and an identification of their boundaries, once I topologically glue the surfaces together there exists a unique conformal structure on my new surface that is compatible with the conformal structures I started with.

See MR1966191 (2005e:30012) Hamilton, D. H.(1MD) Conformal welding. Handbook of complex analysis: geometric function theory, Vol. 1, 137–146, NorthHolland, Amsterdam, 2002. 30C35 and other papers by David Hamilton. 


The gluing does not necessarily exist, neither it is unique when exists, in such generality as stated. (I am not even speaking of what the "boundary" of a Riemann surface could mean in general). Even in the simplest case when two original surfaces are disks, the glueing does not have to exist for an arbitrary homeomorphism. In addition to the articles of Hamilton suggested in the previous message, I recommend the web site of Christopher Bishop, and old papers by Alfred Huber (in German). 


I recommend the article by Radnell and Schippers [1] for quasisymmetric conformal welding of arbitrary Riemann surfaces. That article doesn't do the hard analysis work, but refers to the book [2] (section III.1.4) for the classical conformal welding of disks. References: 

