2017/04/21 by M. Dajczer, Th. Kasioumis, Dajczer, M. +5
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1704.06401
openalex publication_date 2017/04/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \citeharvey and were initially studied by Bryant \citebr. For any dimension and codimension there is an abundance of non-complete examples fully described by Dajczer and Florit \citeDF2 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.