Published Paper
Inserted: 19 feb 2024
Last Updated: 21 jan 2026
Journal: Ann. Sc. Norm. Super. Pisa, Cl. Sci.
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 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.
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