Spherical minimal immersions of the 3–sphere

Abstract.

In 1966 Takahashi [11] proved that a minimal isometric immersion $f:S^m(1) \to S^N(r)$ of round spheres exists iff $r=\sqrt{m/\lambda_p}$, where $\lambda_p$ is the p-th eigenvalue of the Laplacian on S^m in this case, the components of f are spherical harmonics on S^m of order p. This immersion is unique up to congruence on the range and agrees with the generalized Veronese map if m = 2 as was shown in 1967 by Calabi [1]. In 1971 DoCarmo and Wallach [3] proved that the same rigidity holds for p = 2,3. The main aim of their work, however, was to show that, for $m\geq 3$ and $p\geq 4$, unicity fails, and, indeed, the set of (congruence classes of) minimal isometric immersions $f:S^m\to S^N(\sqrt{m/\lambda_p})$ can be parametrized by a moduli space $\mathcal{M}^p_m$ , a compact convex body in a representation space $\mathcal{F}^p_m$ of SO(m + 1) of dimension $\geq 18$. In 1994, the first author [14] determined the exact dimension of the moduli, and with Gauchman [5] in 1996, revealed intricate connections beween the irreducible components of $\mathcal{F}^p_m$ and the geometry of the immersions these components represent. The purpose of the present paper is to provide a complete geometric description of the fine details of the (boundary of the) 18-dimensional space $\mathcal{M}^p_m$, the first nontrivial moduli. This is made possible by several reductions that make use of the splitting $SO(4)=SU(2)\cdot SU(2)'$ as well as rely on the structure of SU(2) equivariant minimal isometric immersions treated in the work of DeTurck and the second author [2] in 1992. The equivariant imbedding theorem [14] asserts that the structure of $\mathcal{M}^p_m$ reappears in the moduli $\mathcal{M}^p_m$ for $m\geq 3$ and $p\geq 4$.