Timeline for Lower bound on injectivity radius at one point implies lower bound on injectivity radius for a closed manifold
Current License: CC BY-SA 4.0
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| Apr 25 at 8:56 | comment | added | user127309 | An explicit lower bound injrad(π)β₯πΆ(π,π·) under the assumption β1<sec<0 is in Buser-Karcher, Gromov's almost flat manifolds, on page 28. | |
| Apr 25 at 6:16 | vote | accept | Xin Qian | ||
| Apr 25 at 5:43 | comment | added | Ian Agol | If $-1<K<0$, then this follows from the generalized Margulis lemma. See Theorem 9.5 doi.org/10.1007/978-1-4684-9159-3 and the Cheeger-Gromov compactness theorem. mathoverflow.net/a/258518/1345 From compactness, one gets that the pinched nonpositively curved manifolds with volume bounded below are compact in the Hausdorff topology, and hence one gets a lower bound on injectivity radius everywhere. If the volume approaches zero, then the max injectivity radius approaches $0$. By 9.5, the fundamental group is virtually nilpotent, and hence not negatively curved, a contradiction. | |
| Apr 25 at 5:25 | answer | added | Ian Agol | timeline score: 5 | |
| Apr 25 at 2:26 | comment | added | Xin Qian | So, if the manifold is negatively pinched, such as $-1<\sec<0$, then this is true? | |
| Apr 25 at 1:38 | comment | added | Ian Agol | I donβt think one can say anything like this without a bound on curvature. I think I can construct counterexample with the curvature going to $-\infty$. | |
| Apr 24 at 23:17 | history | edited | Xin Qian | CC BY-SA 4.0 |
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| Apr 24 at 18:57 | history | asked | Xin Qian | CC BY-SA 4.0 |