Calculus of Variations and Geometric Measure Theory

A. Cucinotta - A. Mondino

Half Space Property in RCD(0,N) spaces

created by mondino on 19 Feb 2024



Inserted: 19 feb 2024
Last Updated: 19 feb 2024

Year: 2024


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.