2006/04/01 by Dmitry Jakobson, Nikolai Nadirashvili, Nikolaï Nadirashvili +1 · 3 citations
Mathematics · #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds
paper · pdf · doi:10.4153/cjm-2006-016-0
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 by the first eigenfunctions. Also, this Klein bottle is a bipolar surface for Lawson's -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 λ 1 on a Klein bottle of a given area. We present numerical evidence and prove the first results towards this conjecture.