In Riemannian geometry the
study of minimal submanifolds has given the most important, higher-dimensional
generalizations of geodesics. Especially significant from a global point of view are
closed minimal submanifolds (generalizing closed geodesies); these raise many hard
problems. In this paper we study existence and uniqueness questions in the case of
the simplest topological type; i.e. minimal hyperspheres. We restrict ourselves to
study such questions for the compact two-point homogeneous spaces; these spaces
constitute the most natural generalization of classical (three-point homogeneous)
spherical geometry. They can be characterized equivalently as (i) compact
two-point homogeneous spaces, (ii) compact rank 1 symmetric spaces, or (iii)
irreducible compact positively curved symmetric spaces. Since the standard
spheres have been investigated in great detail in connection with the “Spherical
Bernstein Problem”, we only consider the complex projective spaces CP(n), the
quaternionic projective spaces HP(n), and the Cayley projective plane Ca(2)
here.
