Preprint
Inserted: 19 feb 2024
Last Updated: 19 feb 2024
Year: 2024
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.
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 result, we also obtain some properties of Green's functions on $RCD(K,N)$ spaces that are of independent interest.
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