Greatest least eigenvalue of the Laplacian on the Klein bottle
| dc.creator | Soufi, Ahmad El | |
| dc.creator | Giacomini, Hector | |
| dc.creator | Jazar, Mustapha | |
| dc.date | 2005-06-29 | |
| dc.date.accessioned | 2026-07-07T05:21:13Z | |
| dc.date.available | 2026-07-07T05:21:13Z | |
| dc.description | We prove the following conjecture recently formulated by Jakobson, Nadirashvili and Polterovich \cite{JNP}: For any Riemannian metric $g$ on the Klein bottle $\mathbb{K}$ one has $$λ\_1 (\mathbb{K}, g) A (\mathbb{K}, g)\le 12 πE(2\sqrt 2/3),$$ where $λ\_1(\mathbb{K},g)$ and $A(\mathbb{K},g)$ stand for the least positive eigenvalue of the Laplacian and the area of $(\mathbb{K},g)$, respectively, and $E$ is the complete elliptic integral of the second kind. Moreover, the equality is uniquely achieved, up to dilatations, by the metric $$g\_0= {9+ (1+8\cos ^2v)^2\over 1+8\cos^2v} (du^2 + {dv^2\over 1+8\cos ^2v}),$$ with $0\le u,v <π$. The proof of this theorem leads us to study a Hamiltonian dynamical system which turns out to be completely integrable by quadratures. | |
| dc.description | 17 pages | |
| dc.identifier | https://arxiv.org/abs/math/0506585 | |
| dc.identifier | http://arxiv.org/abs/math/0506585 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/75609 | |
| dc.subject | Metric Geometry | |
| dc.subject | 58J50; 58E11; 35P15; 37C27 | |
| dc.title | Greatest least eigenvalue of the Laplacian on the Klein bottle | |
| dc.type | text |