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

In this paper we prove general criticality criteria for operators Δ + 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 δ-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 ℝ⁴, we prove that a 1/3-stable minimal hypersurface must either have one end or be a catenoid, and that proper, δ stable minimal hypersurfaces with δ > 1/3 must be hyperplanes.