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Greatest least eigenvalue of the Laplacian on the Klein bottle

Ahmad El Soufi·Hector Giacomini·Mustapha Jazar
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Abstract

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 $$λ\_1 (\mathbb{K}, g) A (\mathbb{K}, g)\le 12 πE(2\sqrt 2/3),$$ where $λ\_1(\mathbb{K},g)$ 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 $$g\_0= {9+ (1+8\cos ^2v)^2\over 1+8\cos^2v} (du^2 + {dv^2\over 1+8\cos ^2v}),$$ with $0\le u,v <π$. The proof of this theorem 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. 2005-06-29. Greatest least eigenvalue of the Laplacian on the Klein bottle. https://arxiv.org/abs/math/0506585

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