Greatest least eigenvalue of the Laplacian on the Klein bottle

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

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 Mathematics Department

We prove the following conjecture recently formulated by Jakobson, Nadirashvili and Polterovich \cite{JNP}: For any Riemannian metric $g$ on the Klein bottle $\mathbb{K}$ one has $$\lambda_1 (\mathbb{K}, g) A (\mathbb{K}, g)\le 12 \pi E(2\sqrt 2/3),$$ where $\lambda_1(\mathbb{K},g)$ and $A(\mathbb{K},g)$ stand for the least positive eigenvalue of the Laplacian and the area of $(\mathbb{K},g)$, respectively, and $E$ is the complete elliptic integral of the second kind. Moreover, the equality is uniquely achieved, up to dilatations, by the metric $$g_0= {9+ (1+8\cos ^2v)^2\over 1+8\cos^2v} \left(du^2 + {dv^2\over 1+8\cos ^2v}\right),$$ with $0\le u,v <\pi$. The proof of this theorem leads us to study a Hamiltonian dynamical system which turns out to be completely integrable by quadratures.

Mots clés: eigenvalue; Laplacian; Klein bottle; Hamiltonian system; integrable system integrable system

Domaines: Géométrie métrique [math.MG]

Ahmad El Soufi, Hector Giacomini, Mustapha Jazar. Greatest least eigenvalue of the Laplacian on the Klein bottle. 2005. ⟨hal-00005692⟩