A splitting theorem for manifolds with a convex boundary component and applications

Journal article

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 [ChodoshLi-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.

Authors: Cucinotta, A; Mondino, A

• Publication status: Accepted
• Peer review status: Peer reviewed
• Publisher: EMS Press
• Journal: Revista Matemática Iberoamericana
• Acceptance date: 2026-06-22
• EISSN: 2235-0616
• ISSN: 0213-2230
• Language: English