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The Cheeger-Gromov compactness theorem says the following. Let us fix $n\in \mathbb{N}$ and positive constants $K,D,v$. Let $\{(M_i^n,g_i)\}$ be a sequence of closed infinitely smooth $n$-dimensional Riemannian manifolds with $|Sec(M_i)|\leq K$, diameter at most $D$ and volume at least $v$. Then, after a choice of a subsequence, there exist a closed smooth $n$-dimensional manifold $M$ with a Riemannian metric $g$ of class $C^{1,\alpha}$ for any $\alpha\in(0,1)$, and diffeomorphisms $\phi_i\colon M\to M_i$ such that $\phi_i^*(g_i)$ converges to $g$ in $C^{1,\alpha}$ for any $\alpha\in (0,1)$.

Question 1. What is a reference for this result in this particular form?

Question 2. I have an impression that I have heard that there are some refinements of the above result saying that the diffeomorphisms $\phi_i$ can be chosen in such a way that the pull-back under them of the Riemann curvature tensor of $g_i$ converges to some tensor in a Sobolev space. Is that correct? What is the precise statement? A reference?

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    $\begingroup$ Anderson proved that the space $\mathcal M(λ,i_0,D)$ of compact Riemannian n-manifolds such that $|Ric|≤λ$, $ inj≥i_0>0$,$ diam≤D$ is compact in the $C^{1,\alpha}$ topology. This result is a generalization of the Cheeger-Gromov compactness theorem . See Anderson, Michael T. Convergence and rigidity of manifolds under Ricci curvature bounds. Invent. Math. 102 (1990), no. 2, 429–445. $\endgroup$ – user21574 Oct 18 '17 at 17:21
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Question 1:

Peters, Stefan Convergence of Riemannian manifolds. Compositio Math. 62 (1987), no. 1, 3–16.

and

Greene, R. E.; Wu, H. Lipschitz convergence of Riemannian manifolds. Pacific J. Math. 131 (1988), no. 1, 119–141.

Question 2:

This follows by the proofs of Peters and Greene-Wu using standard Sobolev estimates, instead of Schauder estimates, for elliptic PDE's (see, for example, the book of Gilbarg-Trudinger).

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  • $\begingroup$ Thank you. Regarding question 2: what statement follows using Sobolev's estimates? What is the precise result? $\endgroup$ – MKO Jan 1 '17 at 19:12
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    $\begingroup$ The metric $g$, written in the appropriate coordinates (namely, harmonic coordinates), would converge in $W^{2,p}$ for any $1 \le p < \infty$. It's the same statement as for a sequence of functions with a uniform bound on the $C^2$ norm. $\endgroup$ – Deane Yang Jan 1 '17 at 21:25

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