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I have a rather straightforward and perhaps somewhat naive question: Is there a Teichmüller theory for open surfaces?

My motivation basically is that I would like to find out more about the "space" of (almost) complex structures on open smooth orientable surfaces $\Sigma$. (For example, I learned the other day from Moishe Kohan the elementary, but nevertheless amusing fact that every orientable smooth surface admits an (almost) complex structure, hence at least two different.)

But from what I've been able to skim through the literature, it appears that most sources focus on the compact case (perhaps for a good reason?).

EDIT: Apologies, I should have also mentioned that I am primarily interested in non-punctured open Riemann surfaces. From skimming the literature I am aware that there are detailed treatments of the punctured case.

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    $\begingroup$ Check Lehto's book "Univalent functions and Teichmuller spaces." He covers Teichmuller spaces of Riemann surfaces of "infinite type". $\endgroup$
    – Moishe Kohan
    Commented Apr 5 at 21:31
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    $\begingroup$ @MoisheKohan: thanks again for yet another helpful tip! I'll check it out. It sounds like what I'm looking for! $\endgroup$
    – M.G.
    Commented Apr 5 at 22:36

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There is a well-developed theory in the punctured case, and a reasonably well-developed theory in the “flaring” case. See here.

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    $\begingroup$ Thanks! I should have mentioned that I am interested in the non-punctured case. Sorry for being dense, but what do you mean by the "flaring" case? $\endgroup$
    – M.G.
    Commented Apr 5 at 20:22
  • $\begingroup$ For example, a Riemann surface minus a round closed disk. The uniformising hyperbolic metric has a “flaring end”. $\endgroup$
    – Sam Nead
    Commented Apr 5 at 21:45
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    $\begingroup$ Thanks, I see now. I was not familiar with that terminology. $\endgroup$
    – M.G.
    Commented Apr 5 at 22:07

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