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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.

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The only rigorous proof of anything like this that I know is in a note that Graeme Segal sent me privately quite a few years ago. At least at that time Graeme seemed to think that it wasn't in the literature, as far as I remember. That's remarkable if true, given that people seem to use this kind of gluing rather freely. The key point is the homotopy invariance of the index of a suitably cooked up family of Fredholm operators. –  Neil Strickland Feb 17 '11 at 8:00
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See MR1966191 (2005e:30012) Hamilton, D. H.(1-MD) Conformal welding. Handbook of complex analysis: geometric function theory, Vol. 1, 137–146, North-Holland, Amsterdam, 2002. 30C35

and other papers by David Hamilton.

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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).

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Could you elaborate on the regularity needed. Does quasisymmetric homeo qualify ? –  BS. Aug 5 '12 at 12:48
    
"Quasisymmetric" is exactly what's needed to have existence and uniqueness. This question of mine (see mathoverflow.net/questions/34763/…) contains a counterexample to uniqueness. –  André Henriques Aug 5 '12 at 19:21
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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:
[1] Radnell and Schippers "Quasisymmetric sewing in rigged Teichmueller space" (http://arxiv.org/abs/math-ph/0507031)
[2] O. Lehto, Univalent functions and Teichm¨ ul ler spaces, Graduate Texts in Mathematics, vol. 109, Springer-Verlag, New York, 1987.

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