# Injectivity radius bounds for Riemannian manifolds of low regularity

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 $$1\leq k<\infty$$?

• 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. Dec 22, 2020 at 7:34
• @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 $C^{\infty}$. Dec 22, 2020 at 9:12
• 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 $r<1$. Dec 23, 2020 at 1:34
• Instead of reference you may say "smooth using Ricci flow, apply the theorem and pass to the limit as time $\to0$". Dec 23, 2020 at 1:40
• 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). Dec 24, 2020 at 7:46