On a stratification of positive scalar curvature compact manifolds

For a compact PSC Riemannian $n$-manifold $(M,g)$, the metric constant $\mathrm {Riem}(g)\in (0, \binom{n}{2}]$ is defined to be the infinimum over $M$ of the spectral scalar curvature $\frac{\sum_{i=1}^N\lambda_i}{\lambda_{\rm max}}$ of $g$, where $\lambda_1, ...,\lambda_N$ are the eigenvalues of the curvature operator of $g$ and $\lambda_{\rm max}$ is the maximal eigenvalue. The functional $g\to \mathrm {Riem}(g)$ is continuous, re-scale invariant and defines a stratification of the space of PSC metrics over $M$. We introduce as well the smooth constant $\mathbf {Riem}(M)\in (0, \binom{n}{2}]$, which is the supremum of $\mathrm {Riem}(g)$ over the set of all psc Riemannian metrics $g$ on $M$. \\ In this paper, we show that in the top layer, compact manifolds with $\mathbf{Riem}=\binom{n}{2}$ are positive space forms. No manifolds have their $\mathbf{Riem}$ in the interval $(\binom{n}{2}-2, \binom{n}{2})$. The manifold $S^{n-1}\times S^1$ and arbitrary connected sums of copies of it with connected sums of positive space forms all have $\mathbf{Riem}=\binom{n-1}{2}$. For $1\leq p\leq n-2\leq 5$, we prove that the manifolds $S^{n-p}\times T^p$ take the intermediate values $\mathbf {Riem}=\binom{n-p}{2}$. From the bottom, we prove that simply connected (resp. $2$-connected, $3$-connected and non-string) compact manifolds of dimension $\geq 5$ (resp. $\geq 7$, $\geq 9$) have $\mathbf{Riem}\geq 1$ (resp. $\geq 3$, $\geq 6$). The proof of these last three results is based on surgery. In fact, we prove that the smooth $\mathbf{Riem}$ constant doesn't decrease after a surgery on the manifold with adequate codimension.