Timeline for Bishop-Gromov volume comparison on manifolds with negligible negative Ricci curvature
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Oct 18, 2017 at 23:37 | answer | added | Anton Petrunin | timeline score: 7 | |
| Oct 18, 2017 at 13:41 | comment | added | Deane Yang | I think this holds but haven't worked out the details. Bishop-Gromov is proved using the Sturm comparison theorem, where the volume form along a geodesic is compared to that of a flat metric. If you simply replace the flat volume form by the solution to $u'' + ku = 0$, where $k$ is a function of distance from the center of a geodesic ball, and becomes $0$ outside some distance $R$, then you get the desired conclusion. | |
| Oct 18, 2017 at 8:58 | comment | added | Mikhail Katz | You should first ask your question for sectional curvature, where I think it is pretty clear that the answer is negative: when a ball encounters a region where the curvature bound does not hold, the volume behavior can change drastically. That's my impression though I am not 100% sure. | |
| Oct 18, 2017 at 8:56 | history | edited | user116108 | CC BY-SA 3.0 |
added 178 characters in body
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| Oct 18, 2017 at 8:54 | comment | added | user116108 | @MikhailKatz By "here", I wanted to mean complete $M$ with compactly supported negative Ricci curvature. Edited the question. | |
| Oct 18, 2017 at 8:53 | comment | added | Mikhail Katz | What does "here" refer to exactly? | |
| Oct 18, 2017 at 8:34 | review | First posts | |||
| Oct 18, 2017 at 8:35 | |||||
| Oct 18, 2017 at 8:30 | history | asked | user116108 | CC BY-SA 3.0 |