Spectral properties of bipolar minimal surfaces in S4S4

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. For the bipolar surface τ˜r,k of the Lawson's torus or Klein bottle τr,kτr,k it is shown that:(1)If rk≡0mod2, τ˜r,k is a torus with an extremal metric for λ4r−2λ4r−2 and λ4r+2λ4r+2.(2)If rk≡1mod4, τ˜r,k is a torus with an extremal metric for λ2r−2λ2r−2 and λ2r+2λ2r+2.(3)If rk≡3mod4, τ˜r,k is a Klein bottle with an extremal metric for λr−2λr−2 and λr+2λr+2. Furthermore, we find explicitly the S1S1-equivariant minimal immersion of the bipolar surfaces into S4S4 by the corresponding eigenfunctions.