Concordance and Isotopy of Metrics with Positive Scalar Curvature

Abstract

Two positive scalar curvature metrics g 0, g 1 on a manifold M are psc-isotopic if they are homotopic through metrics of positive scalar curvature. It is well known that if two metrics g 0, g 1 of positive scalar curvature on a closed compact manifold M are psc-isotopic, then they are psc-concordant: i.e., there exists a metric $${\bar{g}}$$ of positive scalar curvature on the cylinder $${M \times I}$$ which extends the metrics g 0 on $${M \times \{0\}}$$ and g 1 on $${M \times \{1\}}$$ and is a product metric near the boundary. The main result of the paper is that if psc-metrics g 0, g 1 on M are psc-concordant, then there exists a diffeomorphism $${\Phi : M \times I \rightarrow M \times I}$$ with $${\Phi|_{M \times \{0\}} = Id}$$ (a pseudo-isotopy) such that the metrics g 0 and $${(\Phi|_{M \times \{1\}})^{*}g_{1}}$$ are psc-isotopic. In particular, for a simply connected manifold M with dim M ≥ 5, psc-metrics g 0, g 1 are psc-isotopic if and only if they are psc-concordant. To prove these results, we employ a combination of relevant methods: surgery tools related to the Gromov–Lawson construction, classic results on isotopy and pseudo-isotopy of diffeomorphisms, standard geometric analysis related to the conformal Laplacian, and the Ricci flow.