Spectral properties of bipolar minimal surfaces in S^4

Abstract

The i-th eigenvalue of the Laplacian on a surface can be viewed as a functional on the space of Riemannian metrics of fixed area. Extremal points of these functionals correspond to surfaces admitting minimal isometric immersions into spheres. Recently, critical metrics for the first eigenvalue were classified on tori and on Klein bottles. The present paper is concerned with extremal metrics for higher eigenvalues on these surfaces. We apply a classical construction due to Lawson. The ranks of the extremal eigenvalues are obtained for the bipolar surfaces $\tilde \tau_{r,k}$ of the corresponding Lawson's tori or Klein bottles. Furthermore, we find explicitly the $S^1$-equivariant minimal immersion of the bipolar surfaces into $S^4$ by the corresponding eigenfunctions.