On a stratification of positive scalar curvature compact manifolds

For a compact PSC Riemannian [Formula: see text]-manifold [Formula: see text], the metric constant [Formula: see text] is defined to be the infinimum over [Formula: see text] of the spectral scalar curvature [Formula: see text] of [Formula: see text], where [Formula: see text] are the eigenvalues of the curvature operator of [Formula: see text] and [Formula: see text] is the maximal eigenvalue. The functional [Formula: see text] is continuous, re-scale invariant and defines a stratification of the space of PSC metrics over [Formula: see text]. We introduce as well the smooth constant [Formula: see text], which is the supremum of [Formula: see text] over the set of all PSC Riemannian metrics [Formula: see text] on [Formula: see text]. In this paper, we show that in the top layer, compact manifolds with [Formula: see text] are positive space forms. No manifolds have their [Formula: see text] in the interval [Formula: see text]. The manifold [Formula: see text] and arbitrary connected sums of copies of it with connected sums of positive space forms all have [Formula: see text]. For [Formula: see text], we prove that the manifolds [Formula: see text] take the intermediate values [Formula: see text]. From the bottom, we prove that simply connected (resp. [Formula: see text]-connected, [Formula: see text]-connected and non-string) compact manifolds of dimension [Formula: see text] (resp. [Formula: see text], [Formula: see text]) have [Formula: see text] (resp. [Formula: see text], [Formula: see text]). The proof of these last three results is based on surgery. In fact, we prove that the smooth [Formula: see text] constant doesn’t decrease after a surgery on the manifold with adequate codimension.