Contractibility of spaces of positive scalar curvature metrics with symmetry

Abstract

We show the contractibility of spaces of invariant Riemannian metrics of positive scalar curvature on compact connected manifolds of dimension at least two, with and without boundary and equipped with compact Lie group actions. On manifolds without boundary, we assume that the Lie group contains a normal $S^1$-subgroup with fixed-point components of codimension two. In this situation, the existence of invariant metrics of positive scalar curvature was known previously. For the proof, we combine equivariant Morse theory with conformal deformations near unstable manifolds. On manifolds without boundary, we also use local flexibility properties of positive scalar curvature metrics and the smoothing of mean-convex singularities.

36 pages, illustrations and some mathematical details added to improve readability