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.
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!