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arXiv: math.MG/0701773

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Duke Mathematical Journal 135(1) (2006) 181--202

A unique extremal metric for the least eigenvalue of the Laplacian on the Klein bottle

Ahmad El Soufi 1, Hector Giacomini 1, Mustapha Jazar 2

(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

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From: Ahmad El Soufi <>

Submitted on: Friday, 26 January 2007 15:19:13

Updated on: Friday, 26 January 2007 15:49:56