Complete minimal submanifolds with nullity in Euclidean spheres

Abstract

In this paper we investigate m-dimensional complete minimal submanifolds in Euclidean spheres with index of relative nullity at least m−2 at any point. These are austere submanifolds in the sense of Harvey and Lawson [19] and were initially studied by Bryant [3]. For any dimension and codimension there is an abundance of non-complete examples fully described by Dajczer and Florit [7] in terms of a class of surfaces, called elliptic, for which the ellipse of curvature of a certain order is a circle at any point. Under the assumption of completeness, it turns out that any submanifold is either totally geodesic or has dimension three. In the latter case there are plenty of examples, even compact ones. Under the mild assumption that the Omori-Yau maximum principle holds on the manifold, a trivial condition in the compact case, we provide a complete local parametric description of the submanifolds in terms of 1-isotropic surfaces in Euclidean space. These are the minimal surfaces for which the standard ellipse of curvature is a circle at any point. For these surfaces, there exists a Weierstrass type representation that generates all simply connected ones. Let M be a complete m-dimensional Riemannian manifold. In [10] we considered the case of minimal isometric immersions into Euclidean space f : M → R, m ≥ 3, satisfying that the index of relative nullity is at least m − 2 at any point. Under the mild assumption that the Omori-Yau maximum principle holds on M, we concluded that any f must be “trivial”, namely, just a cylinder over a complete minimal surface. This result is global in nature since for any dimension there are plenty of non-complete examples other than open subsets of cylinders. It is natural to expect rather different type of conclusions when considering a similar global problem for minimal isometric immersions into nonflat space forms. For instance, for submanifolds in the hyperbolic space one would guess that under the same condition on the relative nullity index there exist many non-trivial examples, and that a kind of triviality conclusion will only hold under a strong additional assumption. The third author would like to acknowledge financial support from the grant DFG SM 78/6-1.