Preprint

Inserted: 14 jun 2024
Last Updated: 4 jun 2025

Year: 2024

Abstract:

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, and for $4$-manifolds with weakly bounded geometry, non-negative $2$-Ricci curvature, scalar curvature $\geq 1$, and disconnected mean-convex boundary.

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• Cucinotta-Mondino-SplittingConvexBoundary-June2025.pdf