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

Abstract

Let \((\mathcal{M }_3^k)^{SU(2)}\) denote the DoCarmo–Wallach moduli space of \(SU(2)\)- equivariant spherical minimal immersions of the three sphere \(S^3\) of degree \(k\). Although the complexity of these moduli increases rapidly with \(k\) (for example, \(\dim (\mathcal{M }_3^k)^{SU(2)} = \mathcal{O}(k^2)\)), we show here that they possess linear slices that are simplices of dimension \(\mathcal{O}(k)\). 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 Grunbaum type measures of symmetry of these moduli.