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I would appreciate if someone could point me to some introductory literature/resources where I can learn about Poincaré's uniformization theorem at a basic level.

Any good powerpoint notes, short papers or video lectures would be nice. I want to learn about the result in general, the proof, how it relates to other important theorem's in geometry and possible real-life applications.

Thank you.

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On a basic level:

W. Abikoff, The uniformization theorem, Amer. Math. Monthly 88 (1981), no. 8, 574–592.

L. Ahlfors, Conformal invariants, last chapter.

S. Donaldson, Riemann surfaces, Oxford, 2011. Very nice. Modern.

R. Courant, Function theory (if you read German or Russian, this is the second part of the famous old Hurwitz-Courant textbook, not available in English).

On even more basic level:

G. M. Goluzin, Geometric theory of functions of a complex variable, AMS 1969, Appendix.

(It depends on the definition of the Riemann surface that you are willing to accept. If you want to include the triangulability in the definition then Goluzin is fine, and this is probably the simplest proof available. Triangulability is equivalent to the existence of a countable basis of topology, which is not logically necessary to include in the definition (it follows from the modern definition of a RS, but this fact is not trivial). On the other hand, I know of no context where Riemann surfaces arise and the existence of a countable basis is in question. The proofs in Ahlfors and Abikoff include the proof of the existence of a countable basis, and do not discuss triangulability.)

Let me also mention a remarkable recent book:

Henri Paul de Saint-Gervais, Uniformisation des surfaces de Riemann, ENS Éditions, Lyon, 2010. 544 pp. There is an English translation published by EMS.

This is not for quick reading, but it contains a very comprehensive discussion of the history of this theorem, and early attempts to prove it, and various approaches, etc.

EDIT. Let me mention a new book, it is actually a textbook for graduate/undergraduate US students which contains a complete proof:

Donald Marshall, Complex Analysis, Cambridge 2019.

This seems to be the unique general CV textbook which contains a complete proof.

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  • $\begingroup$ I will second Ahlfors Conformal invariants book as the proof there is complete and fairly short and explicit (the whole Conformal Invariants book is a gem having in 150 odd pages material that takes double or more pages in other books and with a very good balance between explanations and brevity) $\endgroup$ – Conrad Jun 17 at 17:56
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Let me second Alex Eremenko's suggestion for

Donald Marshall, Complex Analysis, Cambridge 2019.

The proof is based on the new notion of dipole Green's function, and is especially interesting in view of the following reasons:

  1. It does not require second countability in the definition of a Riemann surface (see Alex's answer), but rather obtain it as a corollary of the uniformization theorem.

  2. It gives existence of meromorphic functions that separate points for arbitrary Riemann surfaces.

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    $\begingroup$ The notion of dipole Green function is not new, see Hurwitz Courant, Nevanlinna, and other early 20 century books. $\endgroup$ – Alexandre Eremenko Jun 17 at 19:23
  • $\begingroup$ @AlexandreEremenko I did not know that! I learned about dipole Green functions from Don Marshall's preprint: sites.math.washington.edu/~marshall/preprints/… which was incorporated in his book. There is no reference to earlier work, so I assumed it was new. Do you know if Hurwitz, Courant, Nevanlinna, etc. proved existence of dipole Green functions on any Riemann surface? $\endgroup$ – Malik Younsi Jun 18 at 20:30
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    $\begingroup$ Nevanlinna did. (In his book Uniformisation) Courant did for comact surfaces, but the proof uses the same idea. Modern people rarely read old books, especialy in German. And all literature on the subject until the middle of 20 century was in German. (I am in a privileged position since many German books have been translated into Russian. Unfortunately my German is poor). $\endgroup$ – Alexandre Eremenko Jun 18 at 23:07
  • $\begingroup$ @AlexandreEremenko Very interesting... Thank you! $\endgroup$ – Malik Younsi Jun 19 at 17:29

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