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

Soufi, Ahmad El; Giacomini, Hector; Jazar, Mustapha

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

Authors

Soufi, Ahmad El

Giacomini, Hector

Jazar, Mustapha

Description

We prove the following conjecture recently formulated by Jakobson, Nadirashvili and Polterovich \cite{JNP}: on the Klein bottle $\mathbb{K}$, the metric of revolution $$g_0= {9+ (1+8\cos ^2v)^2\over 1+8\cos ^2v} (du^2 + {dv^2\over 1+8\cos ^2v}),$$ $0\le u <\fracπ2$, $0\le v <π$, 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.