Criticality, splitting theorems under spectral Ricci bounds and the topology of stable minimal hypersurfaces

In this paper we prove general criticality criteria for operators $\mathrm{\Delta }+V$ on manifolds with more than one end, where $V$ bounds the Ricci curvature, and a related spectral splitting theorem extending Cheeger-Gromoll’s one. Our results give new insight on Li-Wang’s theory of manifolds with a weighted Poincaré inequality. We apply them to study stable and $\delta$-stable minimal hypersurfaces in manifolds with non-negative bi-Ricci or sectional curvature, in ambient dimension up to $5$ and $6$, respectively. In the special case where the ambient space is ${\mathbb{R}}^4$, we prove that a $1\mathrm{/}3$-stable minimal hypersurface must either have one end or be a catenoid, and that proper, $\delta$-stable minimal hypersurfaces with $\delta >1\mathrm{/}3$ must be hyperplanes.