Minimal hyperspheres in two-point homogeneous spaces (Q1919934)

In this paper existence and uniqueness questions for the case of minimal hyperspheres are studied. The compact two-point homogeneous spaces constitute a natural generalization of classical spherical geometry. These spaces can be characterized as (i) compact two-point homogeneous spaces, (ii) compact rank 1 symmetric spaces, or (iii) irreducible compact positively curved symmetric spaces. The complex projective spaces \(\mathbb{C} P(n)\), the quaternionic projective spaces \(\mathbb{H} P(n)\) and the Cayley projective plane \(\text{Ca}(2)\) are investigated. The following results are obtained: (a) there exist infinitely many congruence classes of imbedded minimal hyperspheres in \(\mathbb{C} P(n)\), \(n\geq 2\), and (b) there exist infinitely many congruence classes of immersed, minimal hyperspheres in \(\mathbb{H} P(n)\), \(n\geq 1\), and in the Cayley projective plane Ca(2).