双极Lawson Tau-曲面和广义Lawson Tau-曲面
Bipolar Lawson Tau-Surfaces and Generalized Lawson Tau-Surfaces
摘要：
最近Penskoi [J. Geom. Anal. 25（2015），2645-2666，arXiv：1308.1628]将最小地浸入球体中的劳森τ表面$τ_{r,m}$的众所周知的两参数族推广到最小地浸入球体中的环面和克莱因瓶的三参数族$T_{a,b,c}$。有人指出，这个家庭包括表面进行所有极值度量的第一个非平凡特征值的拉普拉斯-贝尔特拉米算子的环面和克莱因瓶：克利福德环面，等边环面和令人惊讶的双极劳森克莱因瓶$\tildeτ_{3,1}$。在本文中，我们证明了在定理1中，这三参数家庭$T_{a,b,c}$实际上包括所有的双极Lawson τ-表面$\tildeτ_{r,m}$。在定理3中，我们证明了除了$\tildeτ_{3,1}$和等边环面外，广义Lawson曲面上没有度量是最大的。
英文摘要：
Recently Penskoi [J. Geom. Anal. 25 (2015), 2645-2666, arXiv:1308.1628] generalized the well known two-parametric family of Lawson tau-surfaces $τ_{r,m}$ 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 $\tildeτ_{3,1}$. 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 $\tildeτ_{r,m}$. In Theorem 3 we show that no metric on generalized Lawson surfaces is maximal except for $\tildeτ_{3,1}$ and the equilateral torus.
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