Greatest Least Eigenvalue of the Laplacian on the Klein Bottle
نویسندگان
چکیده
We prove the following conjecture recently formulated by Jakobson, Nadirashvili and Polterovich [12]: For any Riemannian metric g on the Klein bottle K one has λ1(K, g)A(K, g) ≤ 12πE(2 √ 2/3), where λ1(K, g) and A(K, g) stand for the least positive eigenvalue of the Laplacian and the area of (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 g0 = 9 + (1 + 8 cos v) 1 + 8 cos v ( du + dv 1 + 8 cos v ) , with 0 ≤ 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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A Unique Extremal Metric for the Least Eigenvalue of the Laplacian on the Klein Bottle
We prove the following conjecture recently formulated by Jakobson, Nadirashvili and Polterovich [15]: on the Klein bottle K, the metric of revolution g0 = 9 + (1 + 8 cos v) 1 + 8 cos v „ du 2 + dv 1 + 8 cos v « , 0 ≤ u < π 2 , 0 ≤ v < π, 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 l...
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