Scalar curvature and volume of a Riemannian manifold (Q1900046)

The author studies the relation between the concepts of volume and of scalar curvature of the Riemannian manifold from the viewpoint of contracting maps. Recall that a map \(f: M\to N\) between Riemannian manifolds of the same dimension \(n\) is said to be \(\varepsilon\)- contracting (on the tangent vectors) if there exists a constant \(\varepsilon>0\) such that \(|f_* (v) |\leq \varepsilon |v|\) for every \(v\in T(M)\). The map \(f\) is \(\varepsilon\)-contracting on \(k\)-planes if \(|f^* (\alpha) |\leq \varepsilon|\alpha|\) for every \(\alpha\in \Lambda^k (T (N))\). As an application counterexamples are given to the use of a weaker hypothesis in the main theorem of a previous paper of the author (entitled ``Sharp estimates and Dirac operator'').