In their seminal paper, Jeff Cheeger, Mikhail Gromov, and Michael Taylor derivated bounds on the injectivity radius of Riemannian manifolds with bounded sectional curvature of the form: $ inj(p)\geq r \frac{Vol(B_M(p,r))}{Vol(B_M(p,r)) + Vol(B_{T_p(M)}(0,2r))} . $ Their paper assumes that the involved manifolds are of $C^{\infty}$-type.

Since then, has anyone derived refinements/extensions for manifolds of lower regularity; i.e.: $C^k$ for some $1\leq k<\infty$?

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    $\begingroup$ Which estimate did you mean? Anyway usually there is no difference --- you can even assume that curvature is bounded from both sides in the sense of Alexandrov. So formally speaking there is no smoothness assumption altho it is $C^{1,1}$ or so. $\endgroup$ Dec 22, 2020 at 7:34
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    $\begingroup$ @AntonPetrunin I added the volume estimate the question. Ah, I figured since I couldn't see a smoothness obstruction below $C^2$ (or I guess $C^{1,1}$) but still the issue is that I would need a reference which states this; and does not set-up in $C^{\infty}$. $\endgroup$ Dec 22, 2020 at 9:12
  • $\begingroup$ Yes, the same proof works in nonsmooth case --- there is no difference. Still, you should specify the curvature bounds --- I assume they are $|sec|\le 1$ + I assume that r is small enuf, say $r<1$. $\endgroup$ Dec 23, 2020 at 1:34
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    $\begingroup$ Instead of reference you may say "smooth using Ricci flow, apply the theorem and pass to the limit as time $\to0$". $\endgroup$ Dec 23, 2020 at 1:40
  • $\begingroup$ Thought, still, a reference would be ideal since my audience are applied mathematicians and not geometers (so its less obvious to them and I don't have space left to re-derive this variant). $\endgroup$ Dec 24, 2020 at 7:46


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