A unique extremal metric for the least eigenvalue of the Laplacian on the Klein bottle
By Ahmad El Soufi, Héctor Giacomini, Mustapha Jazar. 2006
80 citations recorded by the source corpus.
We prove the following conjecture recently formulated by Jakobson, Nadirashvili and Polterovich \cite{JNP}: on the Klein bottle $\mathbb{K}$, the metric of revolution $$g_0= {9+ (1+8\cos ^2v)^2\over 1+8\cos ^2v} \left(du^2 + {dv^2\over 1+8\cos ^2v}\right),$$ $0\le u <\frac\pi 2$, $0\le v <\pi$, 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.
Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/6229404d12d6e5af04dcc6fbe0fe987004367772
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In Applied Mathematics
In Mathematical Physics
Citations in the corpus
- Existence and regularity of maximal metrics for the first Laplace eigenvalue on surfaces (2013, background): “Let us mention the sphere (Hersch [22]), the projective plane (Li-Yau [31]), the torus (Girouard [18] and Nadirashvili [36]), the Klein bottle (El Soufi-Giacomini-Jazar [13] and Jakobson-Nadirashvili-Polterovitch [25]), the genus 2 surfaces (Jakobson-Levitin-Nadirashvili-Nigam-Polterovitch [24], Kar”
- Variational aspects of Laplace eigenvalues on Riemannian surfaces (2011, background): “For the progress on th e subject we refer to the papers [6, 9, 10] as well as [21, 22, 11] and references there.”
- An isoperimetric inequality for Laplace eigenvalues on the sphere (2017, unknown): “It was later shown by El Soufi, Giacomini and Jazar [13] that this metric on ˜ τ 3 , 1 is indeed the unique extremal metric for the functional ¯ λ 1 ( KL , g ) .”