Ricci flow and diffeomorphism groups of 3-manifolds

TL;DR

This paper proves the contractibility of the moduli space of constant curvature metrics on certain 3-manifolds using Ricci flow, completing the proof of the Generalized Smale Conjecture for most cases.

Contribution

It provides a new Ricci flow-based approach to prove the contractibility of the moduli space and completes the proof of the Generalized Smale Conjecture for all but $RP^3$.

Findings

The moduli space of constant curvature metrics is contractible for most spherical and hyperbolic 3-manifolds.

The inclusion of isometry groups into diffeomorphism groups is a homotopy equivalence for these manifolds.

The approach applies uniformly to various space forms and hyperbolic manifolds.

Abstract

We complete the proof of the Generalized Smale Conjecture, apart from the case of $RP^3$, and give a new proof of Gabai's theorem for hyperbolic 3-manifolds. We use an approach based on Ricci flow through singularities, which applies uniformly to spherical space forms other than $S^3$ and $RP^3$ and hyperbolic manifolds, to prove that the moduli space of metrics of constant sectional curvature is contractible. As a corollary, for such a 3-manifold $X$, the inclusion $\text{Isom} (X,g)\to \text{Diff}(X)$ is a homotopy equivalence for any Riemannian metric $g$ of constant sectional curvature.