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.
“…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).…”
Section: Second Variation Of Minimal Lagrangian Surfacessupporting
confidence: 53%
“…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 .…”
We deal with the minimal Lagrangian surfaces of the EinsteinKähler surface S 2 × S 2 , studying local geometric properties and showing that they can be locally described as Gauss maps of minimal surfaces in S 3 ⊂ R 4 . We also discuss the second variation of the area and characterize the most relevant examples by their stability behaviour.
“…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).…”
Section: Second Variation Of Minimal Lagrangian Surfacessupporting
confidence: 53%
“…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 .…”
We deal with the minimal Lagrangian surfaces of the EinsteinKähler surface S 2 × S 2 , studying local geometric properties and showing that they can be locally described as Gauss maps of minimal surfaces in S 3 ⊂ R 4 . We also discuss the second variation of the area and characterize the most relevant examples by their stability behaviour.
“…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.…”
In this short note, we prove that conformal classes which are small perturbations of a product conformal class on a product with a standard sphere admit a metric extremal for some Laplace eigenvalue. As part of the arguments, we obtain perturbed harmonic maps with constant density.
“…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].…”
Abstract. We prove sharp isoperimetric inequalities for Neumann eigenvalues of the Laplacian on triangular domains.The first nonzero Neumann eigenvalue is shown to be maximal for the equilateral triangle among all triangles of given perimeter, and hence among all triangles of given area. Similar results are proved for the harmonic and arithmetic means of the first two nonzero eigenvalues.
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