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Cheeger's Finiteness Theorem states that For each positive numbers $D,v,n$, the number of diffeomorphism classes of Riemannian manifolds $M$ with $Diameter(M)\le D$, $Vol(M)\ge v$, and $|K(M)|\le 1$ is finite. Where $K(M)$ denotes the sectional curvatures of $M$.

Clearly, when a sequence of manifolds ${N^n_k}$, in the class mentioned in Cheegers theorem, converges to $M^n$ in the Gromov-Hausdorff distance, then for $k$ large, $N^n_k$ is diffeomorphic to $M^n$. Is there any estimate of the derivatives of the diffeomorphism constructed in Cheeger's paper? Or is the diffeomorphism converges to idendity in $C^1$ norm?

flag	asked Feb 8 2012 at 21:42 MG 644●4●11

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Yes, the Lipschitz constants converge to 1 and if you make right parametrization then metrics converge in $C^{1,1}$-topology. – Anton Petrunin Feb 8 2012 at 22:53

Thanks Anton. Is there any result when there is only lower bound on curvature assumed? – MG Feb 8 2012 at 23:48

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@MG: under a lower curvature bound there is Perelman's stability theorem, see arxiv.org/abs/math/0703002. Perelman claimed the stability homeomorphisms can be chosen bi-Lipschitz (but I do not think the bi-Lipschitz constants in his proof are supposed to converge to 1). The proof of the claim is unwritten and nobody except Perelman seems to know it. There is of course lots of other finiteness theorems under various assumptions, see e.g. library.msri.org/books/Book30/files/petersen.pdf and referneces therein. – Igor Belegradek Feb 9 2012 at 3:09

@Igor, Thanks. However the stability theorem holds for more general Alexandrov spaces not just manifold. My question is when focusing on manifolds with lower curvature bound only(both the sequence and the limit), is the bi-Lipschitz constants bounded or even converges to 1? or when deal with lowercurvature bound, it is inevitabile to deal with Alexandrov spaces? – MG Feb 9 2012 at 3:37

@MG: If you look at Kapovitch's paper on Perelman's stability, he discusses in section 3 some simplifications in the manifold case coming from controlled topology. I doubt these simplifications help prove Perelman's claim in the manifold case. As for whether it is "inevitable to deal with Alexandrov spaces", I think that avoiding Alexandrov spaces is a wrong strategy, e.g. these spaces appear naturally as limits in studying the collapsing case, and they encode essential geometry of the collapsing sequence. How can you ignore them? The same holds for limits with two-sided curvature bounds. – Igor Belegradek Feb 9 2012 at 12:52

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