Spherical minimal immersions of the 3-sphere

.   In 1966 Takahashi [11] proved that a minimal isometric immersion of round spheres exists iff , where 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 and , unicity fails, and, indeed, the set of (congruence classes of) minimal isometric immersions can be parametrized by a moduli space , a compact convex body in a representation space of SO(m + 1) of dimension . 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 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 , the first nontrivial moduli. This is made possible by several reductions that make use of the splitting 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 reappears in the moduli for and .