Bipolar Lawson Tau-Surfaces and Generalized Lawson Tau-Surfaces

Recently Penskoi [J. Geom. Anal. 25 (2015), 2645–2666, arXiv:1308.1628] generalized the well known two-parametric family of Lawson tau-surfaces τr,m minimally immersed in spheres to a three-parametric family Ta,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 τ̃3,1. In the present paper we show in Theorem 1 that this threeparametric family Ta,b,c includes in fact all bipolar Lawson tau-surfaces τ̃r,m. In Theorem 3 we show that no metric on generalized Lawson surfaces is maximal except for τ̃3,1 and the equilateral torus.