Timeline for Length decreasing homotopies of curves
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 25, 2018 at 13:45 | comment | added | Igor Rivin | @aglearner Yes, I agree, it's not quite there. | |
| Jan 25, 2018 at 9:10 | comment | added | aglearner | Igor, sorry I realized that the confusion probably happened because I had not given the definition of a "length decreasing homotopy". This is a homotopy such that for any $t_1<t_2$ the length of $\varphi_{t_2}(S^1)$ is smaller than the length $\varphi_{t_1}(S^1)$. You agree that in the present form your answer does not settle this question? | |
| Jan 25, 2018 at 2:06 | comment | added | Igor Rivin | @aglearner each curve in the sequence is homotopic to any other one, same is true for any subsequence. | |
| Jan 25, 2018 at 1:23 | comment | added | aglearner | Exactly, and my question is about a continuous homotopy, not about a sequence. I don't see how to get such a homotopy from what you propose | |
| Jan 25, 2018 at 1:15 | comment | added | Igor Rivin | It does not need to converge. There will be a convergent subsequence whose length will go to the infimum. | |
| Jan 25, 2018 at 0:55 | comment | added | aglearner | Igor, thanks. If you are speaking of Brikhoff shortening process, it is known that it does not need to converge: homepages.warwick.ac.uk/~masgak/papers/bhb-catone.pdf so I don't see why this will give you a continuous homotopy. | |
| Jan 25, 2018 at 0:26 | history | answered | Igor Rivin | CC BY-SA 3.0 |