2024/02/19 by Alessandro Cucinotta, Andrea Mondino, Cucinotta, Alessandro +1
Computer Science · Mathematics · #Advanced Operator Algebra Research #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #Digital Image Processing Techniques #FOS: Mathematics #Finite Group Theory Research #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2402.12230
openalex publication_date 2024/02/19 · openalex created_date 2024/02/21 · openalex updated_date 2026/08/04
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 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∈ ℝ and N∈ (1,∞), under the stronger assumption that C is the boundary of a globally perimeter minimizing set. 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), 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.