CMC foliations on Euclidean spaces are minimal foliations

Abstract

In this article, we provide a complete answer to one of the most classical conjectures in differential geometry (of foliations). More precisely, we present a complete positive answer to the classical conjecture, due to Meeks III, Pérez, and Ros (2008), that states that every foliation on ${\mathbb{R}}^{n+1}$ by (possibly varying) CMC hypersurfaces is a foliation by minimal hypersurfaces. Moreover, if $n\le 6$, such a CMC foliation must consist of parallel hyperplanes.

arXiv commentSome typos were fixed. 21 pages