# Elliptic, parabolic and hyperbolic Riemann surfaces: classification?

In the book of Kra and Farkas on Riemann surfaces the following (slightly unusual) definition is given:

Definition IV.3.2 (Section IV.3). Let $$M$$ be a Riemann surface. We will call $$M$$ elliptic if and only if $$M$$ is compact. We will call $$M$$ parabolic if and only if $$M$$ is not compact and $$M$$ doesn't carry a non-negative subharmonic function. We will call $$M$$ hyperbolic if and only if $$M$$ does carry a negative non-constant subharmonic function.

Question. Is there some geometric way to characterise parabolic and hyperbolic surfaces? For example, suppose $$M$$ is a compact Riemann surface and $$x_1,\ldots, x_n$$ are points on it. Is the surface $$M\setminus \{x_1,\ldots, x_n\}$$ parabolic?

• Strange definition, indeed. Usually, compact Riemann surfaces of genus $\geq 2$ are called hyperbolic, because they are uniformized by the upper half-space (or, equivalently, by the disk). – Francesco Polizzi Aug 17 '20 at 10:04
• Should the second occurrence of $M$ in each of the defining sentences be $\widetilde{M}$, the universal cover of $M$? – Michael Albanese Aug 17 '20 at 10:28
• No Michael, I've just copied the definition from the book. And I am certain that this is exactly what they meant. – aglearner Aug 17 '20 at 10:50

This is somewhat unusual terminology, but it is common in the theory of classification of open Riemann surfaces. The more standard notation is $$P_G$$ for "parabolic", and $$O_G$$ for "hyperbolic".
The surface $$M\backslash\{ x_1,\ldots,x_n\}$$ is parabolic in this sense, by the "removable singularity theorem" (a subharmonic function which is bounded from above in a punctured neighborhood of the point extends to a subharmonic function in the full neighborhood).
There are some criteria, especially, for surfaces of the form $$M\backslash E$$, where $$M$$ is compact and $$E$$ is a closed subset. But these criteria are not very geometric: they use capacity. Some results can be given in terms of Hausdorff measures of $$E$$ but they are not "necessary and sufficient".