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We complete the proof of the Generalized Smale Conjecture, apart from the case of R P 3 RP^3 , and give a new proof of Gabai’s theorem for hyperbolic 3 3 -manifolds. We use an approach based on Ricci flow through singularities, which applies uniformly to spherical space forms, except S 3 S^3 and R P 3 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 3 -manifold X X , the inclusion Isom ( X , g ) → Diff ( X ) \operatorname {Isom}(X,g)\rightarrow \operatorname {Diff}(X) is a homotopy equivalence for any Riemannian metric g g of constant sectional curvature.
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Decay of scalar curvature on uniformly contractible manifolds with finite asymptotic dimension
Abstract Gromov proved a quadratic decay inequality of scalar curvature for a class of complete manifolds. In this paper, we prove that for any uniformly contractible manifold with finite asymptotic dimension, its scalar curvature decays to zero at a rate depending only on the contractibility radius of the manifold and the diameter control of the asymptotic dimension. We construct examples of uniformly contractible manifolds with finite asymptotic dimension whose scalar curvature functions decay arbitrarily slowly. This shows that our result is the best possible. We prove our result by studying the index pairing between Dirac operators and compactly supported vector bundles with Lipschitz control. A key technical ingredient for the proof of our main result is a Lipschitz control for the topologicalK‐theory of finite dimensional simplicial complexes.
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- PAR ID:
- 10442256
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Communications on Pure and Applied Mathematics
- Volume:
- 77
- Issue:
- 1
- ISSN:
- 0010-3640
- Format(s):
- Medium: X Size: p. 372-440
- Size(s):
- p. 372-440
- Sponsoring Org:
- National Science Foundation
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