Under spectral Ricci bounds and mean-convex boundary, complete manifolds split isometrically as products or admit positive sectional curvature metrics in dimensions other than 4.
A note on the fundamental group of a compact minimal hypersurface
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Rigidity and flexibility under spectral Ricci lower bounds and mean-convex boundary
Under spectral Ricci bounds and mean-convex boundary, complete manifolds split isometrically as products or admit positive sectional curvature metrics in dimensions other than 4.