Minimal hypersurfaces with zero Gauss-Kronecker curvature

Abstract

We investigate complete minimal hypersurfaces in the Euclidean space ${R}^{4}$, with Gauss-Kronecker curvature identically zero. We prove that, if $f:M^{3}\rightarrow {R}^{4}$ is a complete minimal hypersurface with Gauss-Kronecker curvature identically zero, nowhere vanishing second fundamental form and scalar curvature bounded from below, then $f(M^{3})$ splits as a Euclidean product $L^{2}\times {R}$, where $L^{2}$ is a complete minimal surface in $ {R}^{3}$ with Gaussian curvature bounded from below.