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 (see [15, Conjecture 1.5, page 383]). On the Klein bottle $\mathbb{K}$, the metric of revolution ${\displaystyle g_0=\frac{9+(1+8\cos {}^{2}v){}^2}{1+8\cos {}^{2}v}({\mathrm{du}}^2+\frac{{\mathrm{dv}}^2}{1+8\cos {}^{2}v}),}$ $0\le u<\pi /2$, $0\le v<\pi$, is the 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 that turns out to be completely integrable by quadratures