Riemannian metric for surface of negative Euler characteristic

Question

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)

Answer 1

The Gauss-Bonnet theorem applies equally well in the case of a complete metric of finite area. Therefore negativity of Euler characteristic implies negative sign of curvature.