Minimal hyperspheres in two-point homogeneous spaces

Abstract

In Riemannian geometry the study of minimal submanifolds has given the most important, higher-dimensional generalizations of geodesies. 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.