2024/06/14 by Alessandro Cucinotta, Andrea Mondino, Cucinotta, Alessandro +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2406.09784
openalex publication_date 2024/06/14 · openalex created_date 2024/06/18 · openalex updated_date 2026/07/28
We prove a warped product splitting theorem for manifolds with Ricci curvature bounded from below in the spirit of [Croke-Kleiner, Duke Math. J. (1992)], but instead of asking that one boundary component is compact and mean-convex, we require that it is parabolic and convex. We then deduce several applications, including splitting theorems and first Betti number rigidity results for - 3-manifolds with non-negative Ricci curvature, - 4-manifolds with weakly bounded geometry, non-negative 2-Ricci curvature, scalar curvature ≥ 1. In particular, the latter aswers to a rigidity question posed by [Chodosh-Li-Stryker, JEMS, (2024)]. The proofs rely on a metric gluing of Riemannian manifolds with boundary, resulting in a non-smooth metric space. To address this lack of smoothness, we employ synthetic tools specifically developed for non-smooth settings, with a focus on those based on optimal transportation.