We explain Catino, Mastrolia, and Roncoroni's proof that the hyperplanes are the only orientable stable complete minimal hypersurfaces of $\mathbb{R}^4$.

The study of the second variation of the volume of minimal submanifolds into Riemannian manifolds can be considered as a classical problem in differential geometry. In fact, the operator of the second variation (the Jacobi operator) carries the information about the stability properties of the submanifold when it is thought as a stationary point for the volume functional. We study stable compact minimal submanifolds of the product of a sphere S^m and any Riemannian manifold. This study allows to get the complete classification of the stable compact minimal submanifolds of the product of two spheres, and also, the complete classification of the stable compact minimal surfaces of the product of a 2-sphere and any Riemann surface.