Spherical Minimal Immersions of Spherical Space Forms

Introduction. A number of authors [C], [DW1], [DW2], [L], [T] have studied minimal isometric immersions of Riemannian manifolds into round spheres, and in particular of round spheres into round spheres. As was observed by T. Takahashi [T], if Φ:M → S(r) ⊂ R is such a minimal immersion, then the components of Φ must be eigenfuctions of the Laplace operator on M for the same eigenvalue. And conversely if Φ is an immersion such that all the components are eigenfunctions of the Laplace operator for the same eigenvalue, then Φ is a minimal isometric immersion into a round sphere. Takahashi also observed that ifM is an isotropy irreducible Riemannian homogeneous space, i.e., if the isotropy group of a point acts irreducibly on the tangent space, then an orthonormal basis of each eigenspace automatically gives rise to a minimal isometric immersion into a round sphere. We call these the standard minimal immersions. In particular, if M = S(1) we obtain a sequence of such standard minimal isometric immersions, one for each nonzero eigenvalue. For the first such eigenvalue one obtains the standard embedding into R, and for the