TITLE: Laplace-Beltrami Operator on Surfaces
QUESTION [11 upvotes]: I would like to know what is known about the spectrum of the Laplace-Beltrami operator on 2-dimensional negatively curved surfaces of constant curvature. 
For instance,

What is the spectrum of the Hyperbolic plane of constant curvature $-k$? 
What is the Laplace-Beltrami operator and its spectrum for a compact surface of constant curvature $-1$ and genus $g\geq 2$?
If the surface is compact then the spectrum of the L-B operator is discrete and their eigenvalues are
$$
0=\lambda_1<\lambda_2<\lambda_3<\ldots<\lambda_n<\ldots
$$
In the case of constant curvature $-1$, is there a lower bound for its second eigenvalue? upper bound? what else is known in this case?

Can anyone point me to the right reference? Thanks!

REPLY [6 votes]: If the surface is closed of genus $g$, there is a universal lower bound on $\lambda_{2g-2}$.