On a stratification of positive scalar curvature compact manifolds

For a compact PSC Riemannian $n$-manifold $(M,g)$, the metric constant $\mathrm{R}\mathrm{i}\mathrm{e}\mathrm{m}(g)\in (0,\left(\genfrac{}{}{0px}{}{n}{2}\right)]$ is defined to be the infinimum over $M$ of the spectral scalar curvature $\frac{{\sum }_{i=1}^N{\lambda }_i}{{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}}$ of $g$, where ${\lambda }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\lambda }_N$ are the eigenvalues of the curvature operator of $g$ and ${\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is the maximal eigenvalue. The functional $g\to \mathrm{R}\mathrm{i}\mathrm{e}\mathrm{m}(g)$ is continuous, re-scale invariant and defines a stratification of the space of PSC metrics over $M$. We introduce as well the smooth constant $\mathbf{R}\mathbf{i}\mathbf{e}\mathbf{m}(M)\in (0,\left(\genfrac{}{}{0px}{}{n}{2}\right)]$, which is the supremum of $\mathrm{R}\mathrm{i}\mathrm{e}\mathrm{m}(g)$ over the set of all psc Riemannian metrics $g$ on $M$. \ In this paper, we show that in the top layer, compact manifolds with $\mathbf{R}\mathbf{i}\mathbf{e}\mathbf{m}=\left(\genfrac{}{}{0px}{}{n}{2}\right)$ are positive space forms. No manifolds have their $\mathbf{R}\mathbf{i}\mathbf{e}\mathbf{m}$ in the interval $(\left(\genfrac{}{}{0px}{}{n}{2}\right)-2,\left(\genfrac{}{}{0px}{}{n}{2}\right))$. 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 $\mathbf{R}\mathbf{i}\mathbf{e}\mathbf{m}=\left(\genfrac{}{}{0px}{}{n-1}{2}\right)$. For $1\le p\le n-2\le 5$, we prove that the manifolds $S^{n-p}\times T^p$ take the intermediate values $\mathbf{R}\mathbf{i}\mathbf{e}\mathbf{m}=\left(\genfrac{}{}{0px}{}{n-p}{2}\right)$. From the bottom, we prove that simply connected (resp. $2$-connected, $3$-connected and non-string) compact manifolds of dimension $\ge 5$ (resp. $\ge 7$, $\ge 9$) have $\mathbf{R}\mathbf{i}\mathbf{e}\mathbf{m}\ge 1$ (resp. $\ge 3$, $\ge 6$). The proof of these last three results is based on surgery. In fact, we prove that the smooth $\mathbf{R}\mathbf{i}\mathbf{e}\mathbf{m}$ constant doesn’t decrease after a surgery on the manifold with adequate codimension.