Concordance and isotopy of metrics with positive scalar curvature

الأصل · EN

Two positive scalar curvature metrics g₀, g₁ 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 g₀, g₁ of positive scalar curvature on a closed compact manifold M are psc-isotopic, then they are psc-concordant: i.e., there exists a metric g of positive scalar curvature on the cylinder M× I which extends the metrics g₀ on M× 0 and g₁ on M× 1 and is a product metric near the boundary. The main result of the paper is that if psc-metrics g₀, g₁ on M are psc-concordant, then there exists a diffeomorphism Φ: M× I → M× I with Φ|M× 0=Id (a pseudo-isotopy) such that the metrics g₀ and (Φ|M× 1)*g₁ are psc-isotopic. In particular, for a simply connected manifold M with M≥ 5, psc-metrics g₀, g₁ 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.