Volume growth and positive scalar curvature
The authors verify a conjecture of M. Gromov, namely that a macroscopically large Riemannian manifold cannot have uniformly positive scalar curvature [see \textit{M. Gromov}, Prog. Math. 132, 1-213 (1996; Zbl 0945.53022)], for Riemannian spin manifolds \( M \) with subexponential volume growth. The proof is based on the computation of the higher index of the Dirac operator on \(M\) and on a result of \textit{J. Roe} [J. Differ. Geom. 27, 87-113 (1988; Zbl 0657.58041), Proposition 4.33]. In particular, they prove Gromov's conjecture that a uniformly contractible complete Riemannian manifold \( (M,g)\) (i.e. for every \( r > 0\), there exists \( R>r\) such that every ball with radius \( r \) can be contracted to a point in the ball with radius \(R\)) with bounded geometry (i.e. \((M,g)\) has bounded sectional curvature and positive injectivity radius) cannot have uniformly positive scalar curvature when \((M, g)\) has subexponential volume growth.
uniformly contractible Riemannian manifold
macroscopically large Riemannian manifold
scientific article; zbMATH DE number 1545122