### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 009, 11 pages      arXiv:1406.4652      https://doi.org/10.3842/SIGMA.2016.009

### Bipolar Lawson Tau-Surfaces and Generalized Lawson Tau-Surfaces

Broderick Causley
Department of Mathematics and Statistics, McGill University, Burnside Hall, 805 Sherbrooke Street West, Montreal, QC H3A 0B9, Canada

Received November 27, 2015, in final form January 21, 2016; Published online January 25, 2016

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.

Key words: bipolar surface; Lawson tau-surface; minimal surface; extremal metric.

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