On a stratification of positive scalar curvature compact manifolds

For a compact PSC Riemannian n-manifold (M,g), the metric constant Riem(g)is an element of(0,(n/2)] is defined to be the infinimum over M of the spectral scalar curvature & sum;(N)(i=1)lambda i/ lambda(max) of g, where lambda(1),& mldr;,lambda(N) are the eigenvalues of the curvature operator of gg and lambda(max) is the maximal eigenvalue. The functional g -> Riem(g) is continuous, re-scale invariant and defines a stratification of the space of PSC metrics over MM. We introduce as well the smooth constant Riem(M)is an element of(0,n2], which is the supremum of 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 Riem=n2 are positive space forms. No manifolds have their Riem in the interval ((n2)-2,(n2)). The manifold Sn-1xS1Sn-1xS1 and arbitrary connected sums of copies of it with connected sums of positive space forms all have Riem=n-12. For 1 <= p <= n-2 <= 5, we prove that the manifolds S(n-p)xT(p) take the intermediate values Riem=n-p2. From the bottom, we prove that simply connected (resp. 22-connected, 33-connected and non-string) compact manifolds of dimension >= 5 (resp. >= 7, >= 9) have Riem >= 1 (resp. >= 3, >= 6). The proof of these last three results is based on surgery. In fact, we prove that the smooth RiemRiem constant doesn't decrease after a surgery on the manifold with adequate codimension.