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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.
We investigate complete minimal submanifolds $fcolon M^3toHy^n$ in hyperbolic space with index of relative nullity at least one at any point. The case when the ambient space is either the Euclidean space or the round sphere was already studied in cit
In this paper, we investigate geometric conditions for isometric immersions with positive index of relative nullity to be cylinders. There is an abundance of noncylindrical $n$-dimensional minimal submanifolds with index of relative nullity $n-2$, fu
Let $Sigma$ be a $k$-dimensional complete proper minimal submanifold in the Poincar{e} ball model $B^n$ of hyperbolic geometry. If we consider $Sigma$ as a subset of the unit ball $B^n$ in Euclidean space, we can measure the Euclidean volumes of the
Let $fcolon M^{2n}tomathbb{R}^{2n+ell}$, $n geq 5$, denote a conformal immersion into Euclidean space with codimension $ell$ of a Kaehler manifold of complex dimension $n$ and free of flat points. For codimensions $ell=1,2$ we show that such a subman
E. Cartan proved that conformally flat hypersurfaces in S^{n+1} for n>3 have at most two distinct principal curvatures and locally envelop a one-parameter family of (n-1)-spheres. We prove that the Gauss-Codazzi equation for conformally flat hypersur