Half Space Property in RCD(0,N) spaces

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 ⊂ X × 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. As a consequence, we obtain oscillation estimates and a Half Space Theorem for minimal hypersurfaces in products M × 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.