Timeline for Can scalar curvature and diameter control volume?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Nov 7, 2018 at 14:52 | comment | added | Gabe K | Thanks for the references and for the interesting question. After thinking about it some more, my thoughts got too long for a comment so I posed a follow-up question. mathoverflow.net/questions/314701/… | |
| Nov 7, 2018 at 0:41 | comment | added | Yiyue Zhang | Thanks. For scalar and Ricci curvature, H. Bray proved a theorem in dimension 3, assuming the scalar is larger than n(n-1), Ricci $\ge\epsilon$, he gave a sharp upper bound of the volume. For Ricci and diameter, I only find a paper in 1993: Ricci curvature, diameter and optimal volume bound. Let $M_k=\{M|Ric(M)\ge k(n-1), diam(M)=\pi\}$. $F_k(D)$ is the volume of D-ball in the simple connected space form with the constant sectional curvature k. It was still open that whether $F_k(\pi)$ is the optimal volume bound of $M_k$ in this paper. | |
| S Nov 6, 2018 at 9:55 | history | suggested | Micha Wiedenmann | CC BY-SA 4.0 |
fix typo for -> far
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| Nov 6, 2018 at 9:19 | review | Suggested edits | |||
| S Nov 6, 2018 at 9:55 | |||||
| Nov 6, 2018 at 4:48 | comment | added | Gabe K | As a follow up, to build a counter-example I ended up needing there to be some negative Ricci curvature. I experimented a little bit and couldn't figure out how to build any counter-examples with non-negative Ricci curvature. I wonder if it might be possible to get some volume control with non-negative Ricci, positive scalar and bounded diameter that's better than what the volume comparison theorem states. | |
| Nov 6, 2018 at 4:01 | answer | added | Gabe K | timeline score: 8 | |
| Nov 6, 2018 at 3:21 | answer | added | Anton Petrunin | timeline score: 13 | |
| Nov 6, 2018 at 1:30 | review | First posts | |||
| Nov 6, 2018 at 2:31 | |||||
| Nov 6, 2018 at 1:26 | history | asked | Yiyue Zhang | CC BY-SA 4.0 |