Complete minimal submanifolds with nullity in Euclidean spheres

TL;DR

This paper classifies complete minimal submanifolds with high nullity in Euclidean spheres, showing they are either totally geodesic or three-dimensional, and provides a parametric description using 1-isotropic minimal surfaces.

Contribution

It offers a complete local classification of such submanifolds, linking them to 1-isotropic minimal surfaces and their Weierstrass representations.

Findings

Submanifolds are either totally geodesic or three-dimensional.

Complete examples include compact and non-compact cases.

A Weierstrass type representation for 1-isotropic surfaces is established.

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 \cite{harvey} and were initially studied by Bryant \cite{br}. For any dimension and codimension there is an abundance of non-complete examples fully described by Dajczer and Florit \cite{DF2} 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.