[Paper Review] Ricci flow and contractibility of spaces of metrics

Richard H. Bamler, Bruce Kleiner|arXiv (Cornell University)|Sep 18, 2019

Geometric Analysis and Curvature FlowsMathematics25 references21 citations

TL;DR

This paper establishes that the space of metrics with positive scalar curvature on any closed 3-manifold is either empty or contractible, and that the diffeomorphism group of any 3-dimensional spherical space form deformation retracts onto its isometry group, proving the Generalized Smale Conjecture. The authors use a novel geometric method based on families of singular Ricci flows to construct continuous deformations of metrics, avoiding reliance on prior topological results such as Hatcher’s theorem for $S^3$. The key contribution is a new, flow-based proof of the Smale Conjecture using Ricci flow through singularities.

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.

Motivation & Objective

• To prove that the space of positive scalar curvature (PSC) metrics on any closed 3-manifold is either empty or contractible.
• To establish the Generalized Smale Conjecture by showing that the diffeomorphism group of any 3-dimensional spherical space form deformation retracts onto its isometry group.
• To develop a new geometric method using families of singular Ricci flows to produce continuous deformations of metrics, independent of prior topological techniques.
• To provide a new proof of the Smale Conjecture for $S^3$ that does not rely on Hatcher’s work on the contractibility of embedded 2-spheres in $\mathbb{R}^3$.
• To extend the applicability of Ricci flow techniques to global topological questions about spaces of metrics and diffeomorphism groups.

Proposed method

• The authors construct a rounding process that transforms families of metrics into families of singular Ricci flows with controlled geometry.
• They use a novel technique to produce partial homotopies of families of metrics by evolving them through singular Ricci flows, preserving topological structure.
• A key step involves deforming families of metrics toward conformally flat metrics via a continuous family of flows, leveraging the existence of a continuous family of conformal factors.
• The method relies on the existence and uniqueness of solutions to the Ricci flow with surgery in the presence of singularities, as developed in prior work.
• The authors apply a topological argument based on extending maps from spheres to disks to show that certain metric spaces are contractible.
• They use the fact that the space of conformally flat metrics on a 3-manifold is contractible when equipped with a continuous family of conformal factors, enabling null-homotopies.

Experimental results

Research questions

• RQ1 Is the space of positive scalar curvature metrics on a closed 3-manifold always contractible when non-empty?
• RQ2 Can the Generalized Smale Conjecture be proven using geometric flows rather than classical topology?
• RQ3 Does the diffeomorphism group of a 3-dimensional spherical space form deformation retract onto its isometry group?
• RQ4 Can Ricci flow through singularities be used to construct continuous homotopies of families of metrics?
• RQ5 Is there a flow-based method to prove the Smale Conjecture for $S^3$ that avoids Hatcher’s theorem on embedded spheres?

Key findings

• The space $\operatorname{Met}_{PSC}(M)$ of positive scalar curvature metrics on any closed, orientable 3-manifold $M$ is either empty or contractible.
• The space $\operatorname{Met}_{CC}(M)$ of metrics locally isometric to $S^3$ or $S^2 \times \mathbb{R}$ is either empty or contractible.
• The inclusion of the isometry group into the diffeomorphism group of any 3-dimensional spherical space form is a homotopy equivalence, proving the Generalized Smale Conjecture.
• The proof is independent of Hatcher’s work on the $S^3$ case, providing a new, flow-based proof of the Smale Conjecture.
• The method applies uniformly to all spherical space forms, including $\mathbb{R}P^3$, which was previously excluded in earlier Ricci flow-based proofs.
• The authors establish that the space of spherical structures on $S^2 \times S^1$ is contractible, which implies the homotopy equivalence of the diffeomorphism group and its subgroup preserving the standard structure.