characterization of non compact surfaces in $\mathbb{R^3}$ Is there a way to characterize non compact surfaces with constant mean and gaussian curvature. I know that if $K=0=H$ then the surface is a plane. How can I know about the others? 
Just to add, for compact surfaces with constant positive curvature I know Liebmann's theorem as well. But i want to do it for non compact surfaces.
Any help????
Assuming $K$ and $H$ constant. If $k_1$ and $k_2$ are principal curvature then 
$k_1=H+\sqrt{H^2-K}$ and $k_2=H-\sqrt{H^2-K}$, for compact surfaces I only have $k_1=k_2$ and $k_1>k_2$ is not possible. What can i say about them in this case?

It is a nice exercise that the only surfaces with both principal curvatures constant must be (pieces of) a plane, a sphere, or a right circular cylinder. (As a warm-up, prove that a surface with one constant principal curvature and no umbilic points must be a tube around a regular curve.)
