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Miranda's book on Riemann surfaces ignores the analytical details of proving that compact Riemann surfaces admit nonconstant meromorphic functions, preferring instead to work out the algebraic consequences of (a stronger version of) that assumption. Shafarevich's book on algebraic geometry has this to say:

A harmonic function on a Riemann surface can be conceived as a description of a stationary state of some physical system: a distribution of temperatures, for instance, in case the Riemann surface is a homogeneous heat conductor. Klein (following Riemann) had a very concrete picture in his mind:

"This is easily done by covering the Riemann surface with tin foil... Suppose the poles of a galvanic battery of a given voltage are placed at the points $A_1$ and $A_2$. A current arises whose potential $u$ is single-valued, continuous, and satisfies the equation $\Delta u = 0$ across the entire surface, except for the points $A_1$ and $A_2$, which are discontinuity points of the function."

Does anyone know of a good reference on Riemann surfaces where a complete proof along these physical lines (Shafarevich mentions the theory of elliptic PDEs) is written down? How hard is it to make this appealing physical picture rigorous? (The proof given in Weyl seems too computational and a little old-fashioned. Presumably there are now slick conceptual approaches.)

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up vote 12 down vote accepted

For this and most other things about Riemann surfaces, I recommend Donaldson's Notes on Riemann surfaces, which are based on a graduate course I was once lucky enough to see, and which may eventually make it into book format.

In his account, the "main theorem for compact Riemann surfaces" says that one can solve $\Delta f = \rho$ for any 2-form $\rho$ with integral zero. He describes this as the equation for a steady-state temperature distribution. A full proof is given, but I wouldn't describe it as slick: this is still a substantial result in analysis.

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Ha! I was half way through posting the exact same link, when you posted! I would add that he explains how to use this one equation to deduce the standard results of Riemann surface theory: Hodge decomopsition, Riemann-Roch, uniformisation etc. Incidentally, my understanding is that he hopes to finish the book soon. – Joel Fine Mar 28 '10 at 20:00
Thanks! Donaldson's proof looks much more user-friendly than Weyl's. – Qiaochu Yuan Mar 28 '10 at 20:12

See the book "Introduction to Riemann Surfaces" by G. Springer and references in it.

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The first introductory chapter really is all that is needed for someone who wants to see physical constructions. It uses hydrodynamic constructions largely and starts with a very complex-analysis based approach. I wish I'd encountered it in college. – Justin Curry Mar 29 '10 at 19:55
It is a very good book, a bit old-fashioned, but it is ok. I heard that V.I. Arnold learn the subject using it. – Petya Mar 29 '10 at 20:30

The answers to this previous question are relevant. To fill in some gaps, that question is about building functions on a disc with specified Laplacian. As Tim Perutz says, you want to generalize that to building functions on Riemmann surface whose Laplacian is a specified $2$-form of integral zero.

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For a discrete analogue of the connection between electrical networks and harmonic functions, I suggest taking a look at chapter 9 of the book on Markov Chains and Mixing time by Levin, Peres, and Wilmer.

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The end of the first chapter in McKean and Moll's book on elliptic curves elaborates quite a bit on Klein's picture, (there described in terms of hydrodynamics, rather than electrodynamics). Most of the details are left to the reader, though.

Edit: I guess the bit in the first chapter is more about uniformization than what you're asking. 2.18 discusses differentials of the first kind (those without poles) for compact Riemann surfaces in Klein's hydrodynamical picture. I couldn't find anything about constructing meromorphic functions using this picture, so this may not be so helpful to you.

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