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.