The Schoen–Yau–Gromov–Lawson conjecture on positive scalar curvature and aspherical manifolds

An aspherical manifold is a manifold with contractible universal cover. The Schoen–Yau–Gromov–Lawson conjecture. There is no Riemannian metric of positive scalar curvature on a closed aspherical manifold. This conjecture generalizes the stated obstructions to positive scalar curvature on aspherical manifolds. In dimension three it was proved by Schoen and Yau, and by Gromov and Lawson; the source does not specify the status in higher dimensions.