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.)
Best Answer
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.