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

Abstract : 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.
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Ahmad El Soufi, Hector Giacomini, Mustapha Jazar. A unique extremal metric for the least eigenvalue of the Laplacian on the Klein bottle. Duke Mathematical Journal, Duke University Press, 2006, 135(1), pp.181--202. ⟨hal-00126909⟩

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