Bipolar Lawson Tau-Surfaces and Generalized Lawson Tau-Surfaces

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.

arXiv admin note: text overlap with arXiv:1308.1628 by other authors

References in corpus (7)

• Laplacian eigenvalues functionals and metric deformations on compact manifolds
• Generalized Lawson tori and Klein bottles
• How large can the first eigenvalue be on a surface of genus two?
• Non-maximality of known extremal metrics on torus and Klein bottle
• Finite-gap minimal Lagrangian surfaces in $C{P}^2$
• Maximization of the first nontrivial eigenvalue on the surface of genus two
• Isoperimetric inequality for the third eigenvalue of the Laplace-Beltrami operator on ${\mathbb{S}}^2$