Ricci flow and contractibility of spaces of metrics

TL;DR

This paper proves that the space of positive scalar curvature metrics on 3-manifolds is either empty or contractible and confirms the Generalized Smale Conjecture by showing the deformation retraction of diffeomorphism groups to isometry groups.

Contribution

It provides a new proof of the contractibility of metric spaces and the Smale Conjecture for spherical space forms, independent of previous theorems.

Findings

Space of positive scalar curvature metrics is contractible or empty

Diffeomorphism group deformation retracts to isometry group

New proof of the Smale Conjecture for spherical space forms

Abstract

We show that the space of metrics of positive scalar curvature on any 3-manifold is either empty or contractible. Second, we show that the diffeomorphism group of every 3-dimensional spherical space form deformation retracts to its isometry group. This proves the Generalized Smale Conjecture. Our argument is independent of Hatcher's theorem in the $S^3$ case and in particular it gives a new proof of the $S^3$ case.