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

We prove a warped product splitting theorem for manifolds with Ricci curvature bounded from below in the spirit of [Croke-Kleiner, \emph{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 $\geq 1$.
In particular, the latter aswers to a rigidity question posed by [Chodosh-Li-Stryker, \emph{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.