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

Abstract

Abstract. We prove the following conjecture recently formulated by Jakobson, Nadirashvili and Polterovich [15]: on the Klein bottle K, the metric of revolution, 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 leads us to study a Hamiltonian dynamical system which turns out to be completely integrable by quadratures.

Minimal Lagrangian surfaces in $\Bbb S\sp 2 x $\Bbb S\sp 2$

“…Then we have a minimal immersion Φ: Σ → S 5 ( √ 2) of a Klein bottle Σ such that 1 is the first positive eigenvalue of Δ. From Theorem 1.2 in [12] (take into account the comments in the next paragraph to the statement of the mentioned theorem in [12]), we deduce that our immersion is an embedding and the surface is the Klein bottle B. This proves (3).…”

“…We emphasize an interesting example of a Klein bottle studied in [4,12] whose double cover is an S 1 -equivariant minimal torus in S 4 . Up to congruences, we are going to look at it as a minimal Lagrangian Klein bottle embedded in S 2 × S 2 .…”

Extremal Metrics for Laplace Eigenvalues in Perturbed Conformal Classes on Products

“…denotes the conformal class of a metric g, see for instance [4,5,7,10,11], and references therein. These functionals will not be smooth but only Lipschitz; therefore extremality has to be defined in an appropriate way, see below.…”

Maximizing Neumann fundamental tones of triangles

“…Generalizations of the Szegő-Weinberger result to closed surfaces such as the Klein bottle, the sphere, genus 2 surface, projective plane and equilateral torus are known too [12,19,20,23,27]. For broad surveys of isoperimetric eigenvalue inequalities, one can consult the monographs of Bandle [6], Henrot [18], Kesavan [21] and Pólya-Szegő [32].…”