Mohammed L. Labbi, University of Bahrain

Title: On a stratification of positive scalar curvature compact manifolds

Abstract: Let \( (M,g)\) be a compact Riemannian n-manifold with positive scalar curvature (psc metric). We define a metric constant \( \rm{Riem}(g)\in (0, {n \choose 2})\) to be the infinimum over \( M \) of the spectral scalar curvature \[ \frac{\sum_{i=1}^N\lambda_i}{\lambda_{\rm max}} \] 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 re-scale invariant and defines a stratification of the space of psc metrics over \(M\). We introduce as well the smooth constant \(\mathbf {Riem}(M)\in (0, {n \choose 2})\), which is the supremum of \(\mathrm {Riem}(g)\) over the set of all psc Riemannian metrics \(g\) on \(M\).

In this talk, we show that in the top layer, compact manifolds with \(\mathbf{Riem}(M)={n \choose 2}\) are positive space forms. Then we show that there are no manifolds have their value \(\mathbf {Riem}(M)\) 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 \(1\leq p\leq n-2\leq 5\), 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) compact manifolds of dimension \(\geq 5\) (resp. \(\geq 7\)) have \(\mathbf {Riem}\geq 1\) (resp. \(\geq 3\)). The proof of the two last 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.