CMC foliations on Euclidean spaces are minimal foliations

In this article, we give complete answers to some classical problems and conjectures on differential geometry (of foliations). For instance, we give a complete positive answer to the classical conjecture that states that every foliation on R^n+1 by (possibly varying) CMC hypersurfaces is a foliation by minimal hypersurfaces. Moreover, if n 4 such a CMC foliation must consist of parallel hyperplanes. We prove also that such conjecture holds true in much more general situations, for instance, when the ambient space is a complete Riemannian manifold with non-negative Ricci curvature. We prove also that for a foliation by CMC hypersurfaces on a complete Riemannian manifold M with sectional curvature bounded from below by -K₀ 0, then the mean curvature H of the leaves of the foliation satisfies |H| K₀. This gives a complete positive answer to a conjecture due to Meeks III, P\'erez and Ros. We give some answers to several other problems.