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?

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    $\begingroup$ 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). $\endgroup$ – Francesco Polizzi Aug 17 '20 at 10:04
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    $\begingroup$ Should the second occurrence of $M$ in each of the defining sentences be $\widetilde{M}$, the universal cover of $M$? $\endgroup$ – Michael Albanese Aug 17 '20 at 10:28
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    $\begingroup$ No Michael, I've just copied the definition from the book. And I am certain that this is exactly what they meant. $\endgroup$ – 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".

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.


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