Bipolar Lawson Tau-Surfaces and Generalized Lawson Tau-Surfaces

TL;DR

This paper proves that a three-parameter family of minimal surfaces in spheres encompasses all bipolar Lawson tau-surfaces, and identifies which of these surfaces carry extremal metrics for the first eigenvalue of the Laplace-Beltrami operator.

Contribution

It demonstrates that the family $T_{a,b,c}$ includes all bipolar Lawson tau-surfaces and determines which surfaces have maximal metrics.

Findings

The family $T_{a,b,c}$ includes all bipolar Lawson tau-surfaces.

Only the bipolar Lawson Klein bottle $ ilde{ au}_{3,1}$ and the equilateral torus have maximal metrics.

Most generalized Lawson surfaces do not admit maximal metrics.

Abstract

Recently Penskoi [J. Geom. Anal. 25 (2015), 2645-2666, arXiv:1308.1628] generalized the well known two-parametric family of Lawson tau-surfaces $\tau_{r,m}$ minimally immersed in spheres to a three-parametric family $T_{a,b,c}$ of tori and Klein bottles minimally immersed in spheres. It was remarked that this family includes surfaces carrying all extremal metrics for the first non-trivial eigenvalue of the Laplace-Beltrami operator on the torus and on the Klein bottle: the Clifford torus, the equilateral torus and surprisingly the bipolar Lawson Klein bottle $\tilde{\tau}_{3,1}$. In the present paper we show in Theorem 1 that this three-parametric family $T_{a,b,c}$ includes in fact all bipolar Lawson tau-surfaces $\tilde{\tau}_{r,m}$. In Theorem 3 we show that no metric on generalized Lawson surfaces is maximal except for $\tilde{\tau}_{3,1}$ and the equilateral torus.