Spectral properties of bipolar minimal surfaces in S4

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 (tau) over tilde (r,k) of the Lawson's torus or Klein bottle tau(r,k) it is shown that: (1) If rk equivalent to 0 mod 2, (tau) over tilde (r,k) is a torus with an extremal metric for lambda(4r-2) and lambda(4r+2). (1) If rk equivalent to 1 mod 4, (tau) over tilde (r,k) is a torus with an extremal metric for lambda(2r-2) and lambda(2r+2). (1) If rk equivalent to 3 mod 4, (tau) over tilde (r,k) is a Klein bottle with an extremal metric for lambda(r-2) and lambda(r+2). Furthermore, we find explicitly the S-1-equivariant minimal immersion of the bipolar surfaces into S-4 by the corresponding eigen-functions. (C) 2007 Elsevier B.V. All rights reserved.