Simplicial slices of the space of minimal SU(2)-orbits in spheres

Let (M₃^k)^SU(2) denote the DoCarmo-Wallach moduli space of SU(2)-equivariant spherical minimal immersions of the three sphere S³ of degree k. Although the complexity of these moduli increases rapidly with k (for example, dim (M₃^k) ^SU(2)=O (k²)), we show here that they possess linear slices that are simplices of dimension O(k). The construction of these simplicial slices depend on the DeTurck-Ziller classification of 3-dimensional spherical space forms imbedded into spheres as minimal SU(2)-orbits. The existence of these slices enables us to give asymptotically sharp estimates on a sequence of Grünbaum type measures of symmetry of these moduli.