21732 articles – 15570 references  [version française]
 HAL: hal-00126909, version 1
 arXiv: math.MG/0701773
 Duke Mathematical Journal 135(1) (2006) 181--202
 A unique extremal metric for the least eigenvalue of the Laplacian on the Klein bottle
 (2006)
 We prove the following conjecture recently formulated by Jakobson, Nadirashvili and Polterovich \cite{JNP}: on the Klein bottle $\mathbb{K}$, the metric of revolution $$g_0= {9+ (1+8\cos ^2v)^2\over 1+8\cos ^2v} \left(du^2 + {dv^2\over 1+8\cos ^2v}\right),$$ $0\le u <\frac\pi 2$, $0\le v <\pi$, is the \emph{unique} extremal metric of the first eigenvalue of the Laplacian viewed as a functional on the space of all Riemannian metrics of given area. The proof leads us to study a Hamiltonian dynamical system which turns out to be completely integrable by quadratures.
 1: Laboratoire de Mathématiques et Physique Théorique (LMPT) CNRS : UMR6083 – Université François Rabelais - Tours 2: Mathematics Department Lebanese University
 Subject : Mathematics/Metric Geometry
 Keyword(s): eigenvalue – Laplacian – Klein bottle – extremal metric – Hamiltonian system – integrable system
Attached file list to this document:
 PS
 elsoufi-giaco-jazar1.ps(257.9 KB)
 PDF
 elsoufi-giaco-jazar1.pdf(258 KB)
 hal-00126909, version 1 http://hal.archives-ouvertes.fr/hal-00126909 oai:hal.archives-ouvertes.fr:hal-00126909 From: Ahmad El Soufi <> Submitted on: Friday, 26 January 2007 15:19:13 Updated on: Friday, 26 January 2007 15:49:56