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?