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. The parabolicity assumption cannot be dropped as, otherwise, the catenoid in ambient dimension four would give a counterexample. The convexity assumption, instead, can be relaxed to mean-convexity, if one requires an additional control on the volume growth at infinity. Among the applications, we establish a half-space theorem for mean-convex sets in product manifolds. Additionally, we prove splitting results for: 3-manifolds with non-negative Ricci curvature and disconnected mean-convex boundary, 4-manifolds with weakly bounded geometry, non-negative 2-Ricci curvature, scalar curvature bounded away from zero, and disconnected mean-convex boundary.