A. Cucinotta - A. Mondino
A splitting theorem for manifolds with a convex boundary component and applications to the half-space property
Preprint
Inserted: 14 jun 2024
Last Updated: 19 jul 2024
Year: 2024
Abstract:
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. The parabolicity assumption cannot be dropped as, otherwise, the catenoid in ambient dimension four would give a counterexample. As an application, we deduce a half-space theorem for mean convex sets in product manifolds (resp. for sets whose boundary has mean curvature bounded below by a definite constant, in warped products with negative curvature). The results are obtained by combining glueing techniques for manifolds and optimal transport tools from synthetic Ricci curvature bounds.
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