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?



Elliptic, parabolic and hyperbolic Riemann surfaces: classification?
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".


Classical results can be found in the books


M. Tsuji, Potential theory in modern function theory, Maruzen, Tokyo, 1959 (there is an AMS reprint).


Ahlfors, Sario, Riemann surfaces, Princeton UP, 1960.


