Half Space Property in RCD(K,N) spaces

Abstract

The goal of this note is to prove the Half Space Property for RCD(0,N) spaces, namely that if (X,d,m) is a parabolic RCD(0,N) space and $ C \subset X \times \mathbb{R}$ is locally the boundary of a perimeter minimizing set and it is contained in a half space, then $C$ is a locally finite union of horizontal slices. The same result is proved for RCD(K,N) spaces, for any $K\in \mathbb{R}$ and $N\in (1,\infty)$, under the stronger assumption that $C$ is the boundary of a \emph{globally} perimeter minimizing set. As a consequence, we obtain oscillation estimates and a Half Space Theorem for minimal hypersurfaces in products $M \times \mathbb{R}$, where $M$ is a parabolic smooth manifold (possibly weighted and with boundary), satisfying a Ricci curvature lower bound. On the way of proving the Half Space Property, we also extend to the RCD setting some classical results on Green's functions and parabolic manifolds.