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

Abstract

We prove a warped product splitting theorem for manifolds with Ricci curvature bounded below, in the spirit of Croke–Kleiner [Duke Math. J. 67 (1992)]. Instead of assuming that one boundary component is compact and mean-convex, we require it to be parabolic and convex. We then deduce several applications, including splitting theorems and first Betti number rigidity results for (a) $3$-manifolds with non-negative Ricci curvature, and (b) $4$-manifolds with weakly bounded geometry, non-negative $2$-Ricci curvature, scalar curvature $\ge 1$. In particular, the latter answers a rigidity question posed by Chodosh–Li–Stryker [J. Eur. Math. Soc. 28 (2026)]. 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.