---
tag: RHNJECZ9WR
kind: concept
name: Parabolic and hyperbolic Riemann surfaces
msc: [30F20, 30F15, 31C12]
sources:
  - url: https://en.wikipedia.org/wiki/Riemann_surface
---

## Summary

An open Riemann surface is hyperbolic, in the sense of potential theory, when it carries a nonconstant negative subharmonic function, or equivalently a Green's function.
It is parabolic when it is noncompact and carries no such function.
This splits the open surfaces into those that are "large", like the disc, and those that are "small", like the plane.
On a parabolic surface Brownian motion is recurrent, while on a hyperbolic surface it is transient.

## Explanations

### L1 human

Think of a surface as a sheet with a heat source at one point and the far-away part held at temperature zero.
On a hyperbolic surface heat escapes to infinity well enough that a steady temperature profile exists; this profile is the Green's function.
On a parabolic surface, such as the plane, heat cannot escape fast enough and no such profile exists.
The unit disc is hyperbolic, and the complex plane is parabolic.