Complete stable minimal hypersurfaces in positively curved 4-manifolds

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We show that the combination of non-negative sectional curvature (or 2 -intermediate Ricci curvature) and strict positivity of scalar curvature forces rigidity of complete (non-compact) two-sided stable minimal hypersurfaces in a 4 -manifold with bounded curvature. Our work leads to new comparison results. We also construct various examples showing that rigidity of stable minimal hypersurfaces can fail under other curvature conditions.