Half Space Property in RCD(0,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 locally perimeter minimizing set and it is contained in a half space, then $C$ is a locally finite union of horizontal slices. If the assumption of local perimeter minimizing is strengthened into global perimeter minimizing, then the conclusion can be strengthened into uniqueness of the horizontal slice. 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) with non-negative Ricci curvature. On the way of proving the main results, we also obtain some properties of Green's functions on $RCD(K,N)$ spaces that are of independent interest.