Half Space Property in RCD(K,N) spaces

TL;DR

This paper proves the Half Space Property for RCD(K,N) spaces, showing that perimeter-minimizing boundaries contained in a half space are unions of slices, with applications to minimal hypersurfaces and extensions of classical results.

Contribution

It establishes the Half Space Property for RCD(K,N) spaces, generalizing classical geometric measure theory results to a non-smooth setting.

Findings

Proves the Half Space Property for RCD(0,N) spaces.

Extends the result to RCD(K,N) spaces with global perimeter minimizers.

Derives oscillation estimates and a Half Space Theorem for minimal hypersurfaces.

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.