Generalized Lawson tori and Klein bottles

TL;DR

This paper extends families of minimal tori and Klein bottles in spheres using Takahashi theorem, exploring their spectral extremal properties and introducing a three-parameter family of such surfaces.

Contribution

It introduces a new three-parameter family of minimal tori and Klein bottles in spheres, extending known Lawson surfaces and analyzing their spectral properties.

Findings

Extended Lawson tau-surfaces to three parameters

Identified extremal metrics for eigenvalues on these surfaces

Included metrics extremal for the first non-trivial eigenvalue on tori and Klein bottles

Abstract

Using Takahashi theorem we propose an approach to extend known families of minimal tori in spheres. As an example, the well-known two-parametric family of Lawson tau-surfaces including tori and Klein bottles is extended to a three-parametric family of tori and Klein bottles minimally immersed in spheres. Extremal spectral properties of the metrics on these surfaces are investigated. These metrics include i) both metrics extremal for the first non-trivial eigenvalue on the torus, i.e. the metric on the Clifford torus and the metric on the equilateral torus and ii) the metric maximal for the first non-trivial eigenvalue on the Klein bottle.