Ricci flow and diffeomorphism groups of 3-manifolds

Bamler, Richard; Kleiner, Bruce

doi:10.1090/jams/1003

Ricci flow and diffeomorphism groups of 3-manifolds
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by Richard H. Bamler and Bruce Kleiner

J. Amer. Math. Soc. 36 (2023), 563-589

DOI: https://doi.org/10.1090/jams/1003

Published electronically: August 12, 2022

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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, except $S^3$ and $RP^3$, as well as hyperbolic manifolds, to prove that the space of metrics of constant sectional curvature is contractible. As a corollary, for such a $3$-manifold $X$, the inclusion $\operatorname {Isom}(X,g)\rightarrow \operatorname {Diff}(X)$ is a homotopy equivalence for any Riemannian metric $g$ of constant sectional curvature.

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