On the construction of codimension two minimal immersions of exotic spheres into Euclidean spheres

Abstract

It is, nowadays, a well-known fundamental fact in differential topology that there exist many smooth manifolds which are homeomorphic to but non-diffeomorphic to spheres. Such manifolds were christened exotic spheres by their founder, Milnor [223. The structures of those exotic spheres which bound some parallelizable manifolds were systematically analyzed by the techniques of surgery in [20]. For example, it is not difficult to show that such exotic spheres can be embedded as codimension two submanifolds of the unit spheres. In differential geometry and geometric variational theory, the study of closed minimal submanifolds of the euclidean n-sphere, S"(1), is directly related to that of the local structure of singularities of minimal subvarieties in the general Riemannian setting. Therefore, closed minimal submanifolds of S"(1) are not only interesting, nice, geometric objects by themselves, but are also of general theoretical importance in the study of geometric measure theory [16, 18, 24]. Among various problems on the closed minimal submanifolds of euclidean spheres, the following two problems naturally distinguish themselves as especially interesting, namely: