First eigenvalue of symmetric minimal surfaces in S³

Let λ 1 be the first nontrivial eigenvalue of the Laplacian on a compact surface without boundary. We show that λ 1 = 2 on compact embedded minimal surfaces in S 3 which are invariant under a finite group of reflections and whose fundamental piece is simply connected and has less than six edges. In particular λ 1 = 2 on compact embedded minimal surfaces in S 3 that are constructed by Lawson and by Karcher-Pinkall-Sterling.