On a stratification of positive scalar curvature compact manifolds

TL;DR

This paper introduces a spectral scalar curvature-based stratification of positive scalar curvature manifolds, characterizes the top and intermediate levels, and shows how surgery affects the spectral constant.

Contribution

It defines a new spectral scalar curvature invariant for PSC manifolds, characterizes the extremal and intermediate values, and analyzes its behavior under surgery.

Findings

Manifolds with maximal spectral constant are positive space forms.

No manifolds have spectral constant in the interval (binom(n,2)-2, binom(n,2)).

Surgery preserves or increases the spectral constant under certain conditions.

Abstract

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.