The space of positive scalar curvature metrics on a manifold with boundary

Abstract

We study the space of Riemannian metrics with positive scalar curvature on a compact manifold with boundary. These metrics extend a fixed boundary metric and take a product structure on a collar neighbourhood of the boundary. We show that the weak homotopy type of this space is preserved by certain surgeries on the boundary in co-dimension at least three. Thus, there is a weak homotopy equivalence between the space of such metrics on a simply connected spin manifold W, of dimension n≥6 and with simply connected boundary, and the corresponding space of metrics of positive scalar curvature on the standard disk Dn. Indeed, for certain boundary metrics, this space is weakly homotopy equivalent to the space of all metrics of positive scalar curvature on the standard sphere Sn. Finally, we prove analogous results for the more general space where the boundary metric is left unfixed.