Relationship between two kinds of classifications of Riemann surfaces

Question

There are two kinds of classifications of Riemann surfaces.

Classification 1: Let $M$ be a Riemann surface. We will call $M$:

• elliptic iff $M$ is compact (= closed);
• parabolic iff $M$ is not compact and $M$ doesn't carry a negative non-constant subharmonic function;
• hyperbolic iff $M$ does carry a negative non-constant subharmonic function.

Classification 2: Let $M$ be a Riemann surface. Let $\tilde{M}$ be the universal covering surface of $M$. We'll call $M$:

• elliptic iff $\tilde{M}$ is conformally equivalent to $\mathbb{S}^2$;
• parabolic iff $\tilde{M}$ is conformally equivalent to $\mathbb{C}$;
• hyperbolic iff $\tilde{M}$ is conformally equivalent to the unit disk $\mathbb{D}$.

I want to know, what's the relationship between these two classifications? How to characterize an elliptic/parabolic/hyperbolic surface $M$ in the sense of the first classification by the corresponding Fuchsian group (I mean when $M=\mathbb{D}/\Gamma$ for some Fuchsian group $\Gamma$)?

Answer 1

These classifications coincide for simply connected Riemann surfaces. In general, an open parabolic surface in the second sense is also parabolic in the first sense, but not vice versa. In fact there are only two open parabolic Riemann surfaces in the second sense: the plane and the punctured plane, and they are easily seen to be parabolic in the first sense.

The simplest example of a surface parabolic in the first sense but not in the second sense is a sphere with finitely many $\geq 3$ punctures. For many other examples of parabolic surfaces in the first sense, other than the plane and punctured plane, see, for example, the book of M. Tsuji, Potential theory in modern function theory (Maruzen, Tokyo, 1959, there is an reprint published by Chelsea), Chap. X.

In fact, there are more than two different ways to classify open Riemann surfaces, see Tsuji's book.

Another book which contains a classification open Riemann surfaces is Ahlfors and Sario, Riemann surfaces, and this one is freely available: https://zr9558.files.wordpress.com/2013/11/lars-v-ahlfors-l-sario-riemann-surfaces.pdf