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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 $\Gamma$)?

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  • $\begingroup$ Welcome new contributor. I think you swapped the terms “elliptic” and “parabolic”. Also on your first definition, something is wrong. Did you mean to write that $\tilde{M}$ is compact? $\endgroup$ Aug 20 at 18:56
  • $\begingroup$ The question mathoverflow.net/questions/369379/… has a reference for Classification 1 and an illuminating answer by Eremenko. $\endgroup$ Aug 21 at 13:15
  • $\begingroup$ @JasonStarr The first definition is copied from the book of Farkas and Kra, "Riemann Surfaces 2nd", on page178. I'm not familar with this definition. It seems to be used frequently in geometric analysis. However, in topology, a surface is usually said to be hyperbolic if and only if it carries a hyperbolic metric, which coincides with the second definition. I want to know whether we can characterize a surface of elliptic/parabolic/hyperbolic type in the sense of the first definition by the corresponding Fuchsian group, since it's a common way to define a surface by a Fuchsian group. $\endgroup$
    – gaoqiang
    Aug 21 at 20:19
  • $\begingroup$ @JoeSilverman Thanks. I know this post and it helps a lot. $\endgroup$
    – gaoqiang
    Aug 21 at 20:25

1 Answer 1

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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

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  • $\begingroup$ I didn't read this book. Thank you! $\endgroup$
    – gaoqiang
    Aug 21 at 20:26
  • $\begingroup$ In terms of looking it up, I think the reprint of the Tsuji book was by Chelsea before AMS acquired them (MSN). $\endgroup$
    – LSpice
    Aug 21 at 20:30
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    $\begingroup$ @LSpice: thanks, I did not know this. Corrected. $\endgroup$ Aug 21 at 21:30

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