Due to work of Dajczer and Gromoll in the mid 80's, a non-rigid complete minimal hypersurface $M^n$ in $R^{n+1}$ splits as the euclidean product $M^n=M^3\times R^{n-3}$, where here $M^3$ is a complete minimal hypersurface in $R^4$ with zero Gauss-Kronecker curvature. Hence, the problem reduces in the classification of these 3-dimensional objects. The aim of this talk is to give a complete local classification of minimal hypersurfaces with vanishing Gauss-Kronecker curvature in a 4-dimensional space form. This description is given in terms of the Gauss map of the hypersurface. Moreover, we will give a classification of complete 3-dimensional minimal hypersurfaces with vanishing Gauss-Kronecker curvature and scalar curvature bounded from below.

Based on works by H. Weinberger, R. Hamilton and L. Evans, we obtain a strong elliptic maximum principle for smooth sections in vector bundles over Riemannian manifolds. This maximum principle in the most general form, is sharp and generalizes the classical Hopf strong maximum principle for elliptic operators of second order. We use this maximum principle to give some applications in Geometric Analysis. In particular, we obtain various Bernstein type results for higher co-dimensional graphs generated from maps between Riemannian manifolds. This is a joint work with K. Smoczyk