Minimal SU(2)-orbits in spheres with and without isotropy

Minimal SU(2)-orbits in (unit) spheres of SU(2)-representation spaces form a rich class of spherical minimal submanifolds that have been studied by many authors. In 1992 DeTurck and Ziller showed that all homogeneous spherical 3-space forms can be embedded into spheres as minimal SU(2)-orbits. They also showed that the tetrahedral manifold can be embedded into S⁶ as a minimal SU(2)-orbit, and this not only represents the lowest dimensional non–totally-geodesic example but also is unique (up to isometries of the domain and the range). In our present paper we make a more detailed study of minimal SU(2)-orbits in spheres. We ask which SU(2)-representation spaces admit minimal SU(2)-orbits in their respective unit sphere. We call an SU(2)-representation space receptive if it admits such a minimal orbit. Within a receptive representation space we also ask how many geometrically distinct minimal SU(2)-orbits coexist. Since minimal SU(2)-orbits (and hence receptivity) proliferate in higher dimensions, we impose the condition of isotropy (or helicality), the constancy of the length of the second fundamental form on the unit tangent bundle, on the SU(2)orbits. The main result of this paper is to show that the icosahedral manifold embedded minimally in S¹² represents the lowest dimensional non–totally-geodesic isotropic minimal SU(2)-orbit, but in a striking contrast with the non-isotropic case, it is not unique; there is yet another, geometrically distinct, isotropic minimal SU(2)-orbit of dihedral type not listed in the work of DeTurck and Ziller.