Gromov--Lawson conjecture on positive scalar curvature of aspherical manifolds

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Let $M^n$ be a closed, aspherical manifold, meaning that $\pi_k(M)=0$ for every $k\geq 2$ . Gromov--Lawson conjecture. The manifold $M$ admits no Riemannian metric with positive scalar curvature. This conjecture is stated as an immediate consequence of Gromov's Uryson width conjecture and is also motivated by the filling-radius formulation. In dimension higher than $5$ , the source states that it remains open in general.

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Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

The Gromov–Lawson conjecture on positive scalar curvature of aspherical manifolds
A closed aspherical manifold is a closed manifold whose universal cover is contractible. Gromov–Lawson conjecture. A closed aspherical manifold cannot support a Riemannian metric of positive scalar curvature. This is described as a long-standing conjecture and is used as the model for the paper’s proposed symplectic extension; it remains open in general, although important cases are known.
source: Luca F. Di Cerbo, Alexander Dranishnikov and Ekansh Jauhari, “Symplectically aspherical Kähler manifolds, scalar curvature, and the fundamental group”, arXiv:2607.05170 (2026).

Sources & referencesView supporting material

Primary source

Jinmin Wang, Zhizhang Xie, Guoliang Yu and Bo Zhu, “Filling Radius, Quantitative K-theory and Positive Scalar Curvature”, arXiv:2311.15347 (2024).

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Equivalent formulations 1

The Gromov–Lawson conjecture on positive scalar curvature of aspherical manifolds

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