So, I want to equip a surface of negative Euler characteristic with a Riemannian metric of negative curvature.
I know from the uniformization theorem, that a metric of constant curvature exists
Now, if M is compact (for example a sphere with a finite number > 2 of puncture points):
-I know from Gauss-Bonnet, that this metric has to be of negative curvature
If M is not compact:
-the curvature can't be $> 0$ because the universal covering of M is not the sphere (because the sphere is compact)
Is there a way to rule out the possibility that the metric is of constant curvature = $0$ (flat)