Spectral Properties of Bipolar Minimal Surfaces In

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. For the bipolar surface τ̃r,k of the Lawson’s torus or Klein bottle τr,k it is shown that: (1) If rk ≡ 0 mod 2, τ̃r,k is a torus with an extremal metric for λ4r−2. (2) If rk ≡ 1 mod 4, τ̃r,k is a torus with an extremal metric for λ2r−2. (3) If rk ≡ 3 mod 4, τ̃r,k is a Klein bottle with an extremal metric for λr−2. Furthermore, we find explicitly the S-equivariant minimal immersion of the bipolar surfaces into S by the corresponding eigenfunctions.