Laplace-Beltrami Operator on Surfaces - MathOverflow

Laplace-Beltrami Operator on Surfaces

2011-06-21T13:14:34Z

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!

Answer by Sylvain Bonnot for Laplace-Beltrami Operator on Surfaces

2011-06-21T13:50:15Z

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"

Answer by ε-δ for Laplace-Beltrami Operator on Surfaces

2011-06-21T16:46:21Z

Question #3: No lower bound.

There are metrics on surface of genus 2 (or more) with constant curvature $-1$ and arbitrary big diameter; then diameter goes to infinity, the surface looks more and more like two surfaces with cusps. On such a surface the second eigenvalue can be arbitrary small (if the surface is not connected it is 0).

On the other hand, upper diameter bound gives lower bound on the second eigenvalue.

Answer by Marc Palm for Laplace-Beltrami Operator on Surfaces

2011-06-21T19:38:37Z

I just know about the finite volume case, where the Selberg trace formula is available.

For compact surfaces, the Weyl law tells you that the number of eigenvalues $< T^2$ are approximately $$ \frac{vol(X)}{4 \pi} T^2 + \mathcal{O}_X ( T / \log T).$$

For congruence subgroups, we have a similar formula $$ \frac{vol(X)}{4 \pi} T^2 + c T \log T + \mathcal{O}_X ( T / \log T),$$ and the Selberg Eigenvalue conjecture says that $\lambda_2 \geq 1/4$. Selberg proved $\lambda_2 \geq 3/16$ for congruence subgroups.

Existence: In general, it is not sure wether there are surfaces with only finitely many eigenvalue, if we leave the compact world. On a finite volume surface, the Beltrami Laplace operator admits a continuous spectrum. There are no good bounds available for the contribution of the continuous spectrum in general, which are necessary to show existence of nontrivial eigenvalues via the Selberg trace formula. If you leave the finite volume world, $0$ is not an eigenvalue anymore, but there is a whole book about "Spectra of infinite volume Riemann surfaces".

In the compact world, there exists certain lower and upper bounds in certain situations on certain geometric invariants, but I do not remember the exact reference.

Answer by Agol for Laplace-Beltrami Operator on Surfaces

2011-06-22T01:51:45Z

If the surface is closed of genus $g$, there is a universal lower bound on $\lambda_{2g-2}$.