A splitting theorem for manifolds with a convex boundary component and applications
arXiv:2406.09784
The paper proves a warped‑product splitting theorem for manifolds with a convex, parabolic boundary component under Ricci curvature lower bounds, and uses it to obtain splitting and first Betti number rigidity results for certain 3‑ and 4‑dimensional manifolds via metric gluing and synthetic optimal‑transport tools.
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. We then deduce several applications, including splitting theorems and first Betti number rigidity results for - -manifolds with non-negative Ricci curvature, - -manifolds with weakly bounded geometry, non-negative -Ricci curvature, scalar curvature . 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.
Accepted version. With respect to the previous version, we added a first Betti number rigidity result