Timeline for Length decreasing homotopies of curves
Current License: CC BY-SA 3.0
19 events
| when toggle format | what | by | license | comment | |
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| Sep 22, 2018 at 19:01 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Aug 23, 2018 at 18:02 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Jul 24, 2018 at 18:00 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Jun 24, 2018 at 17:17 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| May 30, 2018 at 22:00 | comment | added | Pietro Majer | A question: Do you take here as length of $\phi \in C^1(\mathbb{S}^1, M) $ its total variation $\int_{\mathbb{S}^1}|\phi'(s)| ds$, or the $\mathcal{H}^1$ measure of $\phi(\mathbb{S}^1)$? (This make a difference for non-injective curves). | |
| May 25, 2018 at 17:04 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Apr 25, 2018 at 17:01 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Mar 26, 2018 at 16:20 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Feb 24, 2018 at 15:30 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Jan 25, 2018 at 19:21 | comment | added | John Pardon | @IgorBelegradek it's just $\mathbb R$. The subscript indicates the name of the variable used as the coordinate of that factor. | |
| Jan 25, 2018 at 19:04 | comment | added | aglearner | Igor, in this example the problem is located close to the geodesic $0\times S^1$, so one can easily make this example compact... | |
| Jan 25, 2018 at 19:01 | comment | added | Igor Belegradek | @JohnPardon: what is $\mathbb R_s$? If it non-compact, that it is ruled out by compactness assumption on $M$. | |
| Jan 25, 2018 at 18:13 | comment | added | aglearner | Thanks a lot John, indeed it looks as a counterexample...I have not thought carefully of what "locally minimizing" means for a geodesic... | |
| Jan 25, 2018 at 17:09 | comment | added | John Pardon | It seems false to me. How about $M=\mathbb R_s\times S^1_t$ with metric $ds^2+(2-e^{-1/s^2}\cos(\frac 1s))dt^2$ as a probable counterexample. The central circle $\{0\}\times S^1$ is not locally length minimizing ($\{\frac 1{2\pi k}\}\times S^1$ has smaller length for any positive integer $k$). I would suspect there is no length decreasing homotopy starting at $\{0\}\times S^1$ (let alone one converging to a locally length minimizing curve). | |
| Jan 25, 2018 at 9:09 | history | edited | aglearner | CC BY-SA 3.0 |
added 145 characters in body
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| Jan 25, 2018 at 0:57 | history | edited | aglearner | CC BY-SA 3.0 |
added 11 characters in body
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| Jan 25, 2018 at 0:26 | answer | added | Igor Rivin | timeline score: 1 | |
| Jan 25, 2018 at 0:19 | comment | added | Igor Rivin | Does a point qualify as locally length-minimizing? | |
| Jan 24, 2018 at 20:18 | history | asked | aglearner | CC BY-SA 3.0 |