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 $\Phi : M\times I \to M\times I$ 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 $\dim M\geq 5$, 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 Gromov-Lawson construction, classic results on isotopy and pseudo-isotopy of diffeomorphisms, standard geometric analysis related to the conformal Laplacian, and the Ricci flow.