# Laplace-Beltrami Operator on Surfaces

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!

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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"

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You might also want to look at Sarnak's Bulletin article: Spectra of Hyperbolic Surfaces. –  Kimball Jun 21 '11 at 14:20
@Sylvain and Kimball: Thanks for the references! –  ght Jun 22 '11 at 0:50

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.

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An for the reference, there is some material in Iwaniec's book "Spectral Methods in Automorphic Forms" Chapter 11, I guess. –  Marc Palm Jun 21 '11 at 19:39
please have also a look at mathoverflow.net/questions/64017/…, and see the first answer for the rigorous statements indicated in my last sentences. –  Marc Palm Jun 21 '11 at 19:56
Thank you for the references! –  ght Jun 22 '11 at 0:50
Correction to my previous comment: I meant "see the accepted answer" –  Marc Palm Jun 22 '11 at 12:47

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

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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.

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