Laplace-Beltrami Operator on Surfaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T02:07:38Z http://mathoverflow.net/feeds/question/68376 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/68376/laplace-beltrami-operator-on-surfaces Laplace-Beltrami Operator on Surfaces ght 2011-06-21T13:14:34Z 2011-06-22T01:51:45Z <p>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. </p> <p>For instance,</p> <ul> <li><p>What is the spectrum of the Hyperbolic plane of constant curvature $-k$? </p></li> <li><p>What is the Laplace-Beltrami operator and its spectrum for a compact surface of constant curvature $-1$ and genus $g\geq 2$?</p></li> <li><p>If the surface is compact then the spectrum of the L-B operator is discrete and their eigenvalues are $$0=\lambda_1&lt;\lambda_2&lt;\lambda_3&lt;\ldots&lt;\lambda_n&lt;\ldots$$ 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?</p></li> </ul> <p>Can anyone point me to the right reference? Thanks!</p> http://mathoverflow.net/questions/68376/laplace-beltrami-operator-on-surfaces/68381#68381 Answer by Sylvain Bonnot for Laplace-Beltrami Operator on Surfaces Sylvain Bonnot 2011-06-21T13:50:15Z 2011-06-21T13:50:15Z <p>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"</p> http://mathoverflow.net/questions/68376/laplace-beltrami-operator-on-surfaces/68407#68407 Answer by ε-δ for Laplace-Beltrami Operator on Surfaces ε-δ 2011-06-21T16:46:21Z 2011-06-21T16:46:21Z <p>Question #3: No lower bound.</p> <p>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).</p> <p>On the other hand, upper diameter bound gives lower bound on the second eigenvalue.</p> http://mathoverflow.net/questions/68376/laplace-beltrami-operator-on-surfaces/68426#68426 Answer by Marc Palm for Laplace-Beltrami Operator on Surfaces Marc Palm 2011-06-21T19:38:37Z 2011-06-21T19:38:37Z <p>I just know about the finite volume case, where the Selberg trace formula is available.</p> <p>For compact surfaces, the Weyl law tells you that the number of eigenvalues $&lt; T^2$ are approximately $$\frac{vol(X)}{4 \pi} T^2 + \mathcal{O}_X ( T / \log T).$$</p> <p>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.</p> <p>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".</p> <p>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. </p> http://mathoverflow.net/questions/68376/laplace-beltrami-operator-on-surfaces/68468#68468 Answer by Agol for Laplace-Beltrami Operator on Surfaces Agol 2011-06-22T01:51:45Z 2011-06-22T01:51:45Z <p>If the surface is closed of genus $g$, there is a <a href="http://www.ams.org/mathscinet-getitem?mr=2560109" rel="nofollow">universal lower bound on $\lambda_{2g-2}$.</a> </p>