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!



Laplace-Beltrami Operator on Surfaces
Probably you should try the following:
Chavel, "Eigenvalues in Riemannian geometry"
Buser "Geometry and spectra of compact Riemann surfaces"
and if you can read french: Berger, Gauduchon, Mazet "Le spectre d'une variete riemannienne"


