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In the Cheeger-Gromov paper "On the Characteristic Numbers of Complete Manifolds of Bounded Curvature and Finite Volume", http://www.maths.ed.ac.uk/~aar/papers/cheegergr1.pdf, the authors make the assertion:

Assertion 0.1. If $M^n$ is complete with sectional curvature $|K|\le 1$, and $Vol(M)<\infty$. Then $M^n$ admits an exhaustion by compact manifolds with smooth boundary, $M^n_k$, such that $Vol(\partial M^n_k)\to 0$ and for which the second fundamental forms $II(\partial M^n_k)$ are uniformly bounded.

As indicated in their paper, the proof depends on a generalization of the argument of Gromov's almost flat manifold paper, which will not be treated in this paper. I am wondering, is there any reference that gives a proof of this assertion?

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Perhaps, ihes.fr/~gromov/PDF/3%5B72%5D.pdf –  Deane Yang Sep 27 '12 at 11:37
The proof in Deane's link is the only one in the literature, which is unfortunate because it is somewhat sketchy. –  Igor Belegradek Sep 27 '12 at 14:34
Igor, I agree. The situation was not satisfactory 20 years ago. I had hoped it would be better by now. I remember wondering if this could be proved by smoothing the distance function from a point using the heat flow and using the level sets of the smoothed function. Does this not work? –  Deane Yang Sep 27 '12 at 14:39
Or maybe rewrite the paper on characteristic numbers in a way that avoids chopping a complete manifold. I tend to believe that there's away to work with smooth compactly supported functions or sections of a vector bundle instead. But I haven't looked at the paper at all, so this comment should be taken with a huge grain of salt. –  Deane Yang Sep 27 '12 at 15:03
Deane: I cannot speak for others. I read the proof long ago and it seemed correct but I would not bet my life on it. There aren't that many applications of the chopping paper, and hence it did not face intense scrutiny. I would certainly advise anyone who wants to use the result to try to go through the proof. –  Igor Belegradek Sep 27 '12 at 16:15
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