Accepted Paper

Inserted: 14 jun 2024
Last Updated: 30 jun 2026

Journal: Revista matemática iberoamericana
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. 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, 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.

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