Bipolar Lawson tau-surfaces and generalized Lawson tau-surfaces

Recently Penskoi 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 2 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 $T_{a,b,c}$ is maximal except for $\tilde{\tau}_{3,1}$ and the equilateral torus.