Extremal Metric for the First Eigenvalue on a Klein Bottle
By Dmitry Jakobson, Nikolai Nadirashvili, Iosif Polterovich. Canadian Journal of Mathematics - Journal Canadien de Mathematiques, 2003
86 citations recorded by the source corpus.
Abstract The first eigenvalue of the Laplacian on a surface can be viewed as a functional on the space of Riemannian metrics of a given area. Critical points of this functional are called extremal metrics. The only known extremal metrics are a round sphere, a standard projective plane, a Clifford torus and an equilateral torus. We construct an extremal metric on a Klein bottle. It is a metric of revolution, admitting a minimal isometric embedding into a sphere ${{\mathbb{S}}^{4}}$ by the first eigenfunctions. Also, this Klein bottle is a bipolar surface for Lawson's ${{\tau }_{3,1}}$ -torus. We conjecture that an extremal metric for the first eigenvalue on a Klein bottle is unique, and hence it provides a sharp upper bound for ${{\lambda }_{1}}$ on a Klein bottle of a given area. We present numerical evidence and prove the first results towards this conjecture.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/2afb56c31d6fb71513d41f5934dd0b412f604bb6
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In Applied Mathematics
In Geometry and Topology
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”
- Laplacian eigenvalue functionals and metric deformations on compact manifolds (2007, background): “For the Klein bottle K, Jacobson, Nadirashvili and Polterovitch [22] showed the existence of a critical metric and El Soufi, Giacomini and Jazar [9] proved that this metric is, up to dilatations, the unique critical metric for λ1 on K.”