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 $$g_0=\frac{9+(1+8{\cos }^{2}v{)}^2}{1+8{\cos }^{2}v}(d{u}^2+\frac{d{v}^2}{1+8{\cos }^{2}v}),$$ $0\le u<\frac{\pi }{2}$, $0\le v<\pi$, 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.