W.Y. Hsiang, W.T. Hsiang and P. Tomter conjectured that every simply-connected, compact symmetric space of dimension ≥4 must contain some minimal hypersurfaces of sphere type. With the aid of equivariant differential geometry, they showed that this is in fact the case for many symmetric spaces of rank one and two. Let M be one of the symmetric spaces: Sn(1)×Sn(1)(n≥4), SU(6)/Sp(3), E6/F4, ℍP2 (quaternionic proj. plane) or CaP2 (Cayley proj. plane). We prove the existence of infmitely many immersed, minimal hypersurfaces of sphere type in M which are invariant under a certain group G of isometries of M. Following Hsiang and the others, the equivariant method is also used here to reduce the problem to an investigation of geodesics in M/G equipped with a metric (with singularities) depending only on the orbital geometry of the transformation group (G, M). However, our constructions are based on area minimizing homogeneous cones, corresponding to a corner singularity of M/G with the local geometry of nodal type; this can be viewed as a variation of some of their constructions which depended on some unstable minimal cones of focal type. We further apply the equivariant method to construct a minimal embedding of S1×Sn−1×Sn−1 into Sn(1)×Sn(1)(n≥2) and a minimal, embedded hypersurface of sphere type in [Formula: see text], ℍPn×ℍPn (n≥2) and CaP2×CaP2.
MINIMAL CONES, PLATEAU'S PROBLEM, AND THE BERNSTEIN CONJECTURE