Bipolar Lawson Tau-Surfaces and Generalized Lawson Tau-Surfaces

Symmetry Integrability and Geometry-Methods and Applications

article

Broderick Causley¹ 

[1] Department of Mathematics and Statistics, McGill University

关键词: bipolar surface;    Lawson tau-surface;    minimal surface;    extremal metric;   

DOI  :  10.3842/SIGMA.2016.009

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown